Homological stability for moduli spaces of high dimensional manifolds. I

Abstract

We prove a homological stability theorem for moduli spaces of simply connected manifolds of dimension $2n > 4$, with respect to forming connected sum with $S^n \times S^n$. This is analogous to Harer’s stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of the moduli spaces of manifolds diffeomorphic to connected sums of $S^n \times S^n$ in a range of degrees.

1. Introduction and statement of results

A famous result of Harer Har85 established homological stability for mapping class groups of oriented surfaces. For example, if $\Gamma _{g,1}$ denotes the group of isotopy classes of diffeomorphisms of an oriented connected surface of genus $g$ with one boundary component, then the natural homomorphism $\Gamma _{g,1} \to \Gamma _{g+1,1}$, given by gluing on a genus one surface with two boundary components, induces an isomorphism in group homology $H_k(\Gamma _{g,1}) \to H_k(\Gamma _{g+1,1})$ as long as $k \leq (2g-2)/3$. (Harer proved this for $k \leq (g+1)/3$, but the range was later improved by Ivanov Iva93 and Boldsen Bol12; see also RW16.) This result can be interpreted in terms of moduli spaces of Riemann surfaces, and it has lead to a wealth of research in topology and algebraic geometry. In this paper we will prove an analogous homological stability result for moduli spaces of manifolds of higher (even) dimension.

Definition 1.1.

For a compact smooth manifold $W$, let $\mathrm{Diff}_\partial (W)$ denote the topological group of diffeomorphisms of $W$ restricting to the identity near its boundary. The moduli space of manifolds of type $W$ is defined as the classifying space $\mathscr{M}(W) = B\mathrm{Diff}_\partial (W)$.

If we are given another compact smooth manifold $W'$ and a codimension zero embedding $W \hookrightarrow W'\setminus \partial W'$, then we obtain a continuous homomorphism $\mathrm{Diff}_\partial (W) \to \mathrm{Diff}_\partial (W')$ by extending diffeomorphisms of $W$ by the identity diffeomorphism on the cobordism $K = W' \setminus \mathrm{int}(W)$. The induced map of classifying spaces shall be denoted

$$\begin{equation} - \cup K : \mathscr{M}(W) \longrightarrow \mathscr{M}(W \cup _{\partial W} K). {\tag{1.1}} \end{equation}$$

We shall give point-set models for $\mathscr{M}(W)$ and the map (1.1) in Section 6.

When $W$ is an orientable surface of genus $g$ with one boundary component, and $W' = W \cup _{\partial W} K$ is also orientable of genus $g+1$ with one boundary component, it can be shown that the map 1.1 is equivalent to the map studied by Harer, and hence it induces an isomorphism on homology in a range of degrees which increases with the genus of the surface. Our main result is analogous to this, but for simply connected manifolds of higher even dimension (although we exclude the case $2n=4$ for the usual reason: we shall need to use the Whitney trick). We must first describe the analogue of genus which we will use.

In each dimension $2n$ we define manifolds

$$\begin{equation*} W_{g,1} = W_{g,1}^{2n} = D^{2n} \sharp g(S^n \times S^n), \end{equation*}$$

the connected sum of $g$ copies of $S^n \times S^n$ with an open disc removed, and if $W$ is a compact path-connected $2n$-manifold, we define the number

$$\begin{equation*} g(W) = \max \{g \in \mathbb{N}\,\,\vert \,\, \text{there exists an embedding $W_{g,1} \hookrightarrow W$}\}, \end{equation*}$$

which we call the genus of $W$. Let $S$ be a manifold obtained by forming the connected sum of $[0,1] \times \partial W$ with $S^n \times S^n$. The corresponding gluing map shall be denoted

$$\begin{equation*} s = - \cup S: \mathscr{M}(W) \longrightarrow \mathscr{M}(W \cup _{\partial W} S). \end{equation*}$$

(If $\partial W$ is not path-connected, then the diffeomorphism type of $S$ relative to $\{0\} \times \partial W$, and hence the homotopy class of $s$, will depend on which path component the connected sum is formed in. The following theorem holds for any such choice.)

Theorem 1.2.

For a simply connected manifold $W$ of dimension $2n \geq 6$, the stabilisation map

$$\begin{equation*} s_*: H_k(\mathscr{M}(W)) \longrightarrow H_k(\mathscr{M}(W \cup _{\partial W} S)) \end{equation*}$$

is an isomorphism if $2k \leq g(W)-3$ and an epimorphism if $2k \leq g(W)-1$.

Our methods are similar to those used to prove many homological stability results for homology of discrete groups, namely to use a suitable action of the group on a simplicial complex. For example, Harer used the action of the mapping class group on the arc complex to prove his homological stability result. In our case the relevant groups are not discrete, so we use a semisimplicial space instead.

1.1. Tangential structures and abelian coefficients

In Section 7 we shall generalise Theorem 1.2 in two directions. First, we shall establish a version of Theorem 1.2 where $\mathscr{M}(W)$ is replaced by a space of manifolds equipped with certain tangential structures; second, we shall allow certain nontrivial systems of local coefficients.

Let $\gamma _{2n} \to BO(2n)$ denote the universal vector bundle. A tangential structure for $2n$-dimensional manifolds is a fibration $\theta : B \to BO(2n)$ with $B$ path-connected, classifying a vector bundle $\theta ^*\gamma _{2n}$ over $B$. Examples include $B = EO(2n)/G \to BO(2n)$ for $G = SO(2n)$, corresponding to an orientation, $G = U(n)$, corresponding to an almost complex structure, or $G = \{1\}$, corresponding to a framing. A $\theta$-structure on a $2n$-dimensional manifold is a map of vector bundles $\hat{\ell }:TW \to \theta ^*\gamma _{2n}$, and we then write $\ell : W \to B$ for the underlying map of spaces. If $\hat{\ell }_{\partial W} : TW\vert _{\partial W} \to \theta ^*\gamma _{2n}$ is a given $\theta$-structure, then we write $\mathrm{Bun}_\partial ^\theta (W;\hat{\ell }_{\partial W})$ for the space of bundle maps extending $\hat{\ell }_{\partial W}$. The group $\mathrm{Diff}_\partial (W)$ acts on this space, and we write

$$\mathscr{M}^\theta (W;\hat{\ell }_{\partial W}) = \mathrm{Bun}_\partial ^\theta (W;\hat{\ell }_{\partial W}) /\!\!/\mathrm{Diff}_\partial (W)$$

for the Borel construction. This space need not be path-connected, but if $\hat{\ell }_W \in \mathrm{Bun}_\partial ^\theta (W;\hat{\ell }_{\partial W})$, then we write $\mathscr{M}^\theta (W,\hat{\ell }_{W})$ for the path component of $\hat{\ell }_W$.

Our generalisation of Theorem 1.2 to manifolds with tangential structures will replace the spaces $\mathscr{M}(W)$ by $\mathscr{M}^\theta (W,\hat{\ell }_{W})$. Before stating the theorem, we must explain the corresponding notion of genus and the analogue of the cobordism $S$.

Definition 1.3.

Let us say that a $\theta$-structure $\hat{\ell } : TW_{1,1} \to \theta ^*\gamma _{2n}$ is admissible if there is a pair of orientation-preserving embeddings $e, f : S^n \times D^n \hookrightarrow W_{1,1} \subset S^n \times S^n$ whose cores $e(S^n \times \{0\})$ and $f(S^n \times \{0\})$ intersect transversely in a single point, such that each of the $\theta$-structures $e^*\hat{\ell }$ and $f^*\hat{\ell }$ on $S^n \times D^n$ extend to $\mathbb{R}^{2n}$ for some orientation-preserving embeddings $S^n \times D^n \hookrightarrow \mathbb{R}^{2n}$. We then define the $\theta$-genus of a compact path-connected $\theta$-manifold $(W, \hat{\ell }_W)$ to be

$$\begin{equation*} g^\theta (W, \hat{\ell }_W) = \max \left\{g \in \mathbb{N}\,\,\bigg |\,\, \text{there are $g$ disjoint copies of $W_{1,1}$ in $(W, \hat{\ell }_W)$, each with admissible $\theta $-structure}\right\}. \end{equation*}$$

If $W$ contains a copy of $W_{g,1}$ such that $\ell _W\vert _{W_{g,1}} : W_{g,1} \to B$ is nullhomotopic, then in Proposition 7.12 we show that $g^\theta (W, \hat{\ell }_W) \geq g-1$, and if $n \neq 3,7$, this can be strengthened to $g^\theta (W, \hat{\ell }_W) \geq g$. When $B$ is simply connected, the number $g^\theta (W,\hat{\ell }_W)$ may be estimated in terms of characteristic numbers, with a constant error term depending only on $n$ and $H_n(B;\mathbb{Z})$; cf. Remark 7.16.

In order to define the stabilisation map, we say that a $\theta$-structure $\hat{\ell }_S$ on $S$ is admissible if it is admissible in the sense above when restricted to $W_{1,1} \subset S$. Suppose furthermore that it restricts to $\hat{\ell }_{\partial W}$ on $\{0\} \times \partial W$, and write $\hat{\ell }_{\partial W}^\prime$ for its restriction to $\{1\} \times \partial W$. Then there is an induced stabilisation map

$$s = - \cup (S, \hat{\ell }_S) : \mathscr{M}^\theta (W,\hat{\ell }_{W}) \longrightarrow \mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{W} \cup \hat{\ell }_S)$$

given by gluing on $S$ to $W$ and extending $\theta$-structures using $\hat{\ell }_S$.

We require two additional terms to describe our result: we say that $\theta$ is spherical if $S^{2n}$ admits a $\theta$-structure, and we say that a local coefficient system is abelian if it has trivial monodromy along all nullhomologous loops.

Theorem 1.4.

For a simply connected manifold $W$ of dimension $2n \geq 6$, a $\theta$-structure $\hat{\ell }_W$ on $W$, an admissible $\theta$-structure $\hat{\ell }_S$ on $S$, and an abelian local coefficient system $\mathcal{L}$ on $\mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{W \cup S})$, the stabilisation map

$$\begin{equation*} s_*: H_k(\mathscr{M}^\theta (W,\hat{\ell }_{W});s^*\mathcal{L}) \longrightarrow H_k(\mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{ W \cup S});\mathcal{L}) \end{equation*}$$

is

(i)

an epimorphism for $3k \leq g^\theta (W, \hat{\ell }_W)-1$ and an isomorphism for $3k \leq g^\theta (W, \hat{\ell }_W)-4$,

(ii)

an epimorphism for $2k \leq g^\theta (W, \hat{\ell }_W)-1$ and an isomorphism for $2k \leq g^\theta (W, \hat{\ell }_W)-3$, if $\theta$ is spherical and $\mathcal{L}$ is constant.

For example, consider the tangential structure $\theta : BU(3) \to BO(6)$. If $(W, \hat{\ell }_W)$ is an almost complex 6-manifold (with non-empty boundary), and $e : W_{g,1} \to W$ is an embedding, then $\ell _W \circ e : W_{g,1} \to BU(3)$ is nullhomotopic because $W_{g,1} \simeq \vee ^{2g} S^3$ and $\pi _3(BU(3))=0$. Thus $g^\theta (W, \hat{\ell }_W) \geq g-1$. Furthermore, $S^6$ admits an almost complex structure so $\theta$ is spherical. So for any admissible $\theta$-structure $\hat{\ell }_S$ on $S$, the stabilisation map

$$s : \mathscr{M}^\theta (W,\hat{\ell }_{W}) \longrightarrow \mathscr{M}^\theta (W \cup _{\partial W} S,\hat{\ell }_{ W \cup S})$$

induces an isomorphism on integral homology in degrees up to $\tfrac{g-4}{2}$.

In the sequel GRW16b to this paper we prove an analogue of Theorem 1.4 where the manifold $S$ is replaced by a more general cobordism $K$, satisfying that $(K,\partial W)$ is $(n-1)$-connected. The theorem proved there includes the case where $W \cup _{\partial W} K$ is a closed manifold.

1.2. Stable homology

If we have a sequence $\hat{\ell }_{S_1}, \hat{\ell }_{S_2}, \ldots$ of admissible $\theta$-structures on $S$ such that $\hat{\ell }_{S_1}\vert _{\{0\} \times \partial W} = \hat{\ell }_W\vert _{\partial W}$ and $\hat{\ell }_{S_i}\vert _{\{1\} \times \partial W} = \hat{\ell }_{S_{i+1}}\vert _{\{0\} \times \partial W}$ for all $i$, then the manifold $W \cup gS$ given by the composition of $W$ and $g$ copies of the cobordism $S$ has a $\theta$-structure $\hat{\ell }_{W \cup gS} = \hat{\ell }_W \cup \hat{\ell }_{S_1} \cup \cdots \cup \hat{\ell }_{S_g}$, and there are maps

$$- \cup (S, \hat{\ell }_{S_{g+1}}) : \mathscr{M}^\theta (W \cup gS, \hat{\ell }_{W \cup gS}) \longrightarrow \mathscr{M}^\theta (W \cup (g+1)S, \hat{\ell }_{W \cup (g+1)S}).$$

In this situation the homology of the limiting space

$$\operatorname *{hocolim}_{g \to \infty } \mathscr{M}^\theta (W \cup gS, \hat{\ell }_{W \cup gS})$$

can be described in homotopy-theoretic terms for any $W$ and any $\theta$, from which explicit calculations are quite feasible. In many cases this description is given in GRW14b, and we shall describe the general case in GRW16b, Sections 1.2 and 7. Here we focus on the interesting special case $W = D^{2n}$ and $\theta = \mathrm{Id}_{BO(2n)}$, in which case $\mathscr{M}^\theta (W \cup gS , \hat{\ell }_{W \cup gS}) = \mathscr{M}(W_{g,1}) = B\mathrm{Diff}_\partial (W_{g,1})$.

The boundary of $W_{g,1}$ is a sphere, so $S = ([0,1] \times S^{2n-1}) \sharp (S^n \times S^n)$, and hence there is a diffeomorphism $W_{g,1} \cup _{\partial W_{g,1}} S \approx W_{g+1,1}$ relative to their already identified boundaries. Theorem 1.2 (or Theorem 1.4 for abelian coefficients) therefore implies the following.

Corollary 1.5.

For $2n \geq 6$ and an abelian coefficient system $\mathcal{L}$ on $\mathscr{M}(W_{g+1,1})$, the stabilisation map

$$\begin{equation*} s_* : H_k(\mathscr{M}(W_{g,1});s^*\mathcal{L}) \longrightarrow H_k(\mathscr{M}(W_{g+1,1});\mathcal{L}) \end{equation*}$$

is an epimorphism for $3k \leq g-1$ and an isomorphism for $3k \leq g-4$. If $\mathcal{L}$ is constant, then it is an epimorphism for $2k \leq g-1$ and an isomorphism for $2k \leq g-3$.

It is an immediate consequence of this corollary and RW13, Section 3 that the mapping class group $\pi _1(\mathscr{M}(W_{g,1}))$ has perfect commutator subgroup as long as $g \geq 7$.

Remark 1.6.

Independently, Berglund and Madsen BM13 have obtained a result similar to this corollary, for rational cohomology in the range $2k \leq \min (2n-6, g-6)$.

Remark 1.7.

In the earlier preprint GRW12 we considered only the manifolds $W_{g,1}$, rather than the more general manifolds of Theorem 1.2. Although the present paper entirely subsumes GRW12, the reader mainly interested in Corollary 1.5 may want to consult the preprint for a streamlined text adapted to that special case (at least if they are only concerned with constant coefficients).

By the universal coefficient theorem, stability for homology implies stability for cohomology; in the surface case, Mumford Mum83 conjectured an explicit formula for the stable rational cohomology, which in our notation asserts that a certain ring homomorphism

$$\begin{equation*} \mathbb{Q}[\kappa _1, \kappa _2, \dots ] \longrightarrow H^*(\mathscr{M}(W^2_{g,1});\mathbb{Q}) \end{equation*}$$

is an isomorphism for $g \gg *$. Mumford’s conjecture was proved in a strengthened form by Madsen and Weiss MW07.

Corollary 1.5 and our previous paper GRW14b allow us to prove results analogous to Mumford’s conjecture and the Madsen–Weiss theorem for the moduli spaces $\mathscr{M}(W_{g,1})$ with $2n \geq 6$. The analogue of the Madsen–Weiss theorem for these spaces concerns the homology of the limiting space $\mathscr{M}(W_\infty ) = \operatorname *{hocolim}\limits _{g \to \infty } \mathscr{M}(W_{g,1})$. There is a certain infinite loop space $\Omega ^\infty MT\theta ^n$ associated to the tangential structure $\theta ^n : BO(2n)\langle n \rangle \to BO(2n)$ given by the $n$-connected cover, and a map

$$\alpha : \mathscr{M}(W_\infty ) \longrightarrow \Omega ^\infty MT\theta ^n$$

given by a parametrised form of the Pontryagin–Thom construction. In GRW14b, Theorem 1.1 we proved that $\alpha$ induces an isomorphism between the homology of $\mathscr{M}(W_\infty )$ and the homology of the basepoint component of $\Omega ^\infty MT\theta ^n$. In GRW16 we used this to compute $H_1(\mathscr{M}(W_{g,1});\mathbb{Z})$ for $g \geq 5$.

It is easy to calculate the rational cohomology ring of a component of $\Omega ^\infty MT\theta ^n$, and hence of $\mathscr{M}(W_{g,1})$ in the range of degrees given by Corollary 1.5. The result is Corollary 1.8 below, which is a higher-dimensional analogue of Mumford’s conjecture. Recall that for each $c \in H^{k+2n}(BSO(2n))$ there is an associated cohomology class $\kappa _c \in H^{k}(\Omega ^\infty MT\theta ^n)$. Pulling it back via $\alpha$ and all the stabilisation maps $\mathscr{M}(W_{g,1}) \to \mathscr{M}(W_{\infty })$ defines classes $\kappa _c \in H^k(\mathscr{M}(W_{g,1}))$ for all $g$, sometimes called “generalised Mumford–Morita–Miller classes”. These can also be described in terms of fibrewise integration; see e.g. GRW14a, Section 1.1. The following result is our higher-dimensional analogue of Mumford’s conjecture.

Corollary 1.8.

Let $2n \geq 6$, and let $\mathcal{B}\subset H^*(BSO(2n);\mathbb{Q})$ be the set of monomials in the classes $e, p_{n-1}, \dots , p_{\lceil \frac{n+1}{4} \rceil }$, of degree greater than $2n$. Then the induced map

$$\begin{equation*} \mathbb{Q}[\kappa _c \,\,|\,\, c \in \mathcal{B}] \longrightarrow H^*(\mathscr{M}(W_{g,1});\mathbb{Q}) \end{equation*}$$

is an isomorphism in degrees satisfying $2* \leq g-3$.

For example, if $2n = 6$, the set $\mathcal{B}$ consists of monomials in $e$, $p_1$, and $p_2$, and therefore $H^*(\mathscr{M}(W_{g,1}^{6});\mathbb{Q})$ agrees for $\ast \leq \tfrac{g-3}{2}$ with a polynomial ring in variables of degrees 2, 2, 4, 6, 6, 6, 8, 8, 10, 10, 10, 10, 12, 12, ….

2. Techniques

In this section we collect the technical results needed to establish high connectivity of certain simplicial spaces which will be relevant for the proof of Theorem 1.2. The main results are Theorem 2.4 and Corollary 2.9.

2.1. Cohen–Macaulay complexes

Recall from HW10, Definition 3.4 that a simplicial complex $X$ is called weakly Cohen–Macaulay of dimension $n$ if it is $(n-1)$-connected and the link of any $p$-simplex is $(n-p-2)$-connected. In this case, we write $\mathrm{wCM}(X) \geq n$. We shall also say that $X$ is locally weakly Cohen–Macaulay of dimension $n$ if the link of any $p$-simplex is $(n-p-2)$-connected (but no global connectivity is required on $X$ itself). In this case we shall write $\mathrm{lCM}(X) \geq n$.

Lemma 2.1.

If $\mathrm{lCM}(X) \geq n$ and $\sigma < X$ is a $p$-simplex, then $\mathrm{wCM}(\mathrm{Lk}(\sigma )) \geq n-p-1$.

Proof.

By assumption, $\mathrm{Lk}(\sigma )$ is $((n-p-1)-1)$-connected. If $\tau < \mathrm{Lk}(\sigma )$ is a $q$-simplex, then

$$\begin{equation*} \mathrm{Lk}_{\mathrm{Lk}(\sigma )}(\tau ) = \mathrm{Lk}_X(\sigma \ast \tau ) \end{equation*}$$

is $((n-p-1) - q - 2)$-connected, since $\sigma \ast \tau$ is a $(p+q+1)$-simplex, and hence its link in $X$ is $(n-(p+q+1)-2)$-connected.

■

Definition 2.2.

Let us say that a simplicial map $f: X \to Y$ of simplicial complexes is simplexwise injective if its restriction to each simplex of $X$ is injective; i.e. the image of any $p$-simplex of $X$ is a (non-degenerate) $p$-simplex of $Y$.

Lemma 2.3.

Let $f: X \to Y$ be a simplicial map between simplicial complexes. Then the following conditions are equivalent:

(i)

$f$ is simplexwise injective,

(ii)

$f(\mathrm{Lk}(\sigma )) \subset \mathrm{Lk}(f(\sigma ))$ for all simplices $\sigma < X$,

(iii)

$f(\mathrm{Lk}(v)) \subset \mathrm{Lk}(f(v))$ for all vertices $v \in X$,

(iv)

the image of any 1-simplex in $X$ is a (non-degenerate) 1-simplex in $Y$.

Proof.

(i) $\Rightarrow$ (ii). If $\sigma = \{v_0, \dots , v_p\}$ and $v \in \mathrm{Lk}(\sigma )$, then $\{v,v_0, \dots , v_p\} < X$ is a simplex, and therefore $\{f(v), f(v_0), \dots , f(v_p)\} < Y$ is a simplex. Since $f$ is simplexwise injective, we must have $f(v) \not \in f(\sigma )$, so $f(v) \in \mathrm{Lk}(f(\sigma ))$.

(ii) $\Rightarrow$ (iii). Trivial.

(iii) $\Rightarrow$ (iv). If $\sigma = \{v_0, v_1\}$ is a 1-simplex, then $v_1 \in \mathrm{Lk}(v_0)$ so $f(v_1) \in \mathrm{Lk}(f(v_0))$, but then $\{f(v_0), f(v_1)\}$ is a 1-simplex.

(iv) $\Rightarrow$ (i). Let $\sigma = \{v_0, \dots , v_p\} < X$ be a $p$-simplex and assume for contradiction that $f\vert _\sigma$ is not injective. This means that $f(v_i) = f(v_j)$ for some $i \neq j$, but then the restriction of $f$ to the 1-simplex $\{v_i, v_j\}$ is not injective.

■

The following theorem generalises the “colouring lemma” of Hatcher and Wahl HW10, Lemma 3.1, which is the special case where $X$ is a simplex. The proof given below is an adaptation of theirs.

Theorem 2.4.

Let $X$ be a simplicial complex with $\mathrm{lCM}(X) \geq n$, $f: \partial I^n \to |X|$ be a map, and $h : I^n \to \vert X \vert$ be a nullhomotopy. If $f$ is simplicial with respect to a piecewise linear (PL) triangulation $\partial I^n \approx |L|$, then this triangulation extends to a PL triangulation $I^n \approx |K|$ and $h$ is homotopic relative to $\partial I^n$ to a simplicial map $g : |K| \to |X|$ such that

(i)

for each vertex $v \in K \setminus L$, the star $v *\mathrm{Lk}_K(v)$ intersects $L$ in a single (possibly empty) simplex, and

(ii)

for each vertex $v \in K \setminus L$, $g(\mathrm{Lk}_K(v)) \subset \mathrm{Lk}_X(g(v))$.

In particular, $g$ is simplexwise injective if $f$ is.

Proof.

We proceed by induction on $n$, the case $n=0$ being clear. By the simplicial approximation theorem we may change the map $h$ by a homotopy relative to the boundary, after which $h: I^n \to |X|$ is simplicial with respect to some PL triangulation $I^n \approx |K|$ extending the given triangulation on $\partial I^n$. After barycentrically subdividing $K$ relative to $L$ twice, (i) is satisfied.

Let us say that a simplex $\sigma < K$ is bad if every vertex $v \in \sigma$ is contained in a 1-simplex $(v,v') \subset \sigma$ with $h(v) = h(v')$. If all bad simplices are contained in $\partial I^n$, we are done. If not, let $\sigma < K$ be a bad simplex not contained in $\partial I^n$, of maximal dimension $p$. We may then write $I^n = A \cup B$ for subsets

$$\begin{align*} A &= \sigma * \mathrm{Lk}_K(\sigma ),\\ B &= I^n \setminus \mathrm{int}(A),\\ A \cap B &= (\partial \sigma ) * \mathrm{Lk}_K(\sigma ), \end{align*}$$

which are also subcomplexes with respect to the triangulation $I^n \approx |K|$. We shall describe a procedure which changes the triangulation of $A$ as well as the map $h\vert _{A}$ in a way that strictly decreases the number of bad simplices of dimension $p$ and not contained in $\partial I^n$, creates no bad simplices of higher dimension, and preserves the property (i). This will complete the proof, since we may eliminate all bad simplices not contained in $\partial I^n$ in finitely many steps.

Badness of $\sigma$ implies $p > 0$, and we must also have $h(\mathrm{Lk}_K(\sigma )) \subset \mathrm{Lk}_X(h(\sigma ))$, since otherwise we could join a vertex of $\mathrm{Lk}_K(\sigma )$ to $\sigma$ and get a bad simplex of strictly larger dimension. Now $|\sigma | \subset |K| \approx I^n$, so $h$ restricts to a map

$$\begin{equation*} \partial I^{n-p} \approx |\mathrm{Lk}_K(\sigma )| \longrightarrow |\mathrm{Lk}_X(h(\sigma ))|. \end{equation*}$$

The image $h(\sigma )$ is a simplex of dimension at most $p-1$, since otherwise $h|_\sigma$ would be injective (in fact $h(\sigma )$ must have dimension at most $(p-1)/2$ by badness). Then $|\mathrm{Lk}_X(h(\sigma ))|$ is $(n-(p-1)-2)$-connected since we assumed $\mathrm{lCM}(X) \geq n$, and in fact Lemma 2.1 gives

$$\begin{equation*} \mathrm{wCM}(\mathrm{Lk}_X(h(\sigma ))) \geq n - (p-1) - 1 = n - p. \end{equation*}$$

Therefore, $h|_{|\mathrm{Lk}_K(\sigma )|}$ extends to a map

$$\begin{equation*} \tilde{h} : C(|\mathrm{Lk}_K(\sigma )|) \longrightarrow |\mathrm{Lk}_X(h(\sigma ))|, \end{equation*}$$

and we may apply the induction hypothesis to this map. It follows that there is a PL triangulation $C(|\mathrm{Lk}_K(\sigma )|) \approx \vert \tilde{K} \vert$ satisfying (i) and extending the triangulation of $|\mathrm{Lk}_K(\sigma )|$, and a simplicial map $\tilde{g} : \tilde{K} \to \mathrm{Lk}_X(h(\sigma ))$ extending $h\vert _{\mathrm{Lk}_K(\sigma )}$ and satisfying (ii). In particular, the star of each vertex in $\tilde{K} \setminus \mathrm{Lk}_K(\sigma )$ intersects $\mathrm{Lk}_K(\sigma )$ in a single simplex, and all bad simplices of $\tilde{g}$ are in $\mathrm{Lk}_K(\sigma ) \subset \tilde{K}$ (but in fact there cannot be any, by maximality of $\sigma$). The composition

$$\begin{equation*} A = |\sigma \ast \mathrm{Lk}_K(\sigma )| \approx |(\partial \sigma ) \ast (C \mathrm{Lk}_K(\sigma ))| \xrightarrow {(h|_{\partial \sigma }) * \tilde{g}} |h(\partial \sigma ) * \mathrm{Lk}_X(h(\sigma ))| \subset |X| \end{equation*}$$

agrees with $h$ on $\partial A = (\partial \sigma ) * \mathrm{Lk}_K(\sigma )$ and may therefore be glued to $h\vert _{B}$ to obtain a new map $h': I^n \to |X|$ which is simplicial with respect to a new triangulation $I^n \approx |K'|$ which on $B \subset I^n$ agrees with the old one and on $A$ comes from the triangulation $|\partial \sigma | \ast C(|\mathrm{Lk}_K(\sigma )|) \approx |(\partial \sigma ) \ast \tilde{K}|$. In particular all vertices of $K$ are also vertices of $K'$, and we call these the “old” vertices.

We claim that this new triangulation of $I^n$ still satisfies (i). There are two cases to consider depending on whether $v \in A \setminus \partial A$ or $v \in B$, and we first consider the case where $v \in A \setminus \partial A$. Such a vertex is necessarily interior to the triangulation $C(|\mathrm{Lk}_K(\sigma )|) \approx |\tilde{K}|$, and hence its star intersects $\mathrm{Lk}_K(\sigma )$ in a single simplex $\tau < \mathrm{Lk}_K(\sigma )$ and hence intersects $\partial A = A \cap B \approx |(\partial \sigma ) * \mathrm{Lk}_K(\sigma )|$ in $|(\partial \sigma ) * \tau |$. The intersection of $|L|$ and $|\sigma * \mathrm{Lk}_K(\sigma )|$ is also a single simplex (since the intersection of $L$ and the star of a vertex of $\sigma$ is), of the form $\iota _0 * \iota _1 < (\partial \sigma ) * \mathrm{Lk}_K(\sigma )$. Hence the intersection of $L$ and the star of $v \in K'$ is $\iota _0 *(\iota _1 \cap \tau )$ which is again a simplex.

In the second case where $v \in K'$ is in $B$, it was also a vertex in the old triangulation of $I^n$ by $K$, and its link in $K'$ consists of old vertices which were in its link in $K$, along with some new vertices which are interior to $A$ and hence not in $\partial I^n$, but no old vertices which were not in the link of $v$ in $K$ (since that would violate condition (i) for the triangulation $C(|\mathrm{Lk}_K(\sigma )|) \approx |\tilde{K}|$). Hence the star of $v \in K$ will intersect $L$ in a face of its old intersection, which is still a single simplex.

■

Proposition 2.5.

Let $X$ be a simplicial complex and $Y \subset X$ be a full subcomplex. Let $n$ be an integer with the property that for each $p$-simplex $\sigma < X$ having no vertex in $Y$, the complex $Y \cap \mathrm{Lk}_X(\sigma )$ is $(n-p-1)$-connected. Then the inclusion $|Y| \hookrightarrow |X|$ is $n$-connected.

Proof.

This is very similar to the proof of Theorem 2.4. Let $k \leq n$ and consider a map $h: (I^k, \partial I^k) \to (|X|,|Y|)$ which is simplicial with respect to some PL triangulation of $I^k$. Let $\sigma < I^k$ be a $p$-simplex such that $h(\sigma ) \subset |X|\setminus |Y|$. If $p$ is maximal with this property, we will have $h(\mathrm{Lk}(\sigma )) \subset |Y| \cap \mathrm{Lk}_X(h(\sigma ))$, since otherwise we could make $p$ larger by joining $\sigma$ with a vertex $v \in \mathrm{Lk}(\sigma )$ such that $h(v) \not \in Y$ or $h(v) \in h(\sigma )$.

Now, $\mathrm{Lk}(\sigma ) \approx S^{k-p-1}$ since $\sigma$ is a $p$-simplex, and $|Y| \cap \mathrm{Lk}_X(h(\sigma ))$ is assumed $(n-p-1)$-connected, so $h\vert _{\mathrm{Lk}(\sigma )}$ extends over the cone of $\mathrm{Lk}(\sigma )$. Then modify $h$ on the ball

$$\begin{equation*} \sigma \ast \mathrm{Lk}(\sigma ) \approx (\partial \sigma ) \ast C \mathrm{Lk}(\sigma ) \subset I^k \end{equation*}$$

by replacing it with the join of $h \vert _{\partial \sigma }$ and some map $C \mathrm{Lk}(\sigma ) \to |Y| \cap \mathrm{Lk}_X(h(\sigma ))$ extending $h\vert _{\mathrm{Lk}(\sigma )}$. As in the proof of Theorem 2.4, the modified map $(I^k, \partial I^k) \to (|X|,|Y|)$ is homotopic to the old one (on the ball where the modification takes place, both maps have images in the contractible set $h(\sigma ) \ast \mathrm{Lk}_X(h(\sigma ))$) and has strictly fewer $p$-simplices mapping to $|X|\setminus |Y|$.

■

2.2. Serre microfibrations

Let us recall from Wei05 that a map $p: E \to B$ is called a Serre microfibration if for any $k$ and any commutative diagram

$$\begin{equation*} \vcenter{\img[][84pt][55pt][{$\xymatrix{ \{0\} \times D^k \ar[r]^-f \ar[d] & E \ar[d]^p\\ [0,1] \times D^k \ar[r]^-h & B}$}]{Images/img547b89f3ba053d676aa09fec0e0a330d.svg}} \end{equation*}$$

there exists an $\varepsilon > 0$ and a map $H: [0,\varepsilon ]\times D^k \to E$ with $H(0,x) = f(x)$ and $p \circ H(t,x) = h(t,x)$ for all $x \in D^k$ and $t \in [0,\varepsilon ]$.

Examples of Serre microfibrations include Serre fibrations and submersions of manifolds; if $p: E \to B$ is a Serre microfibration, then $p \vert _U: U \to B$ is too, for any open $U \subset E$.

The microfibration condition implies that if $(X,A)$ is a finite CW-pair, then any map $X \to B$ may be lifted in a neighbourhood of $A$, extending any prescribed lift over $A$. It also implies the following useful observation: suppose $(Y,X)$ is a finite CW-pair and we are given a lifting problem

$$\begin{equation} \begin{aligned} \vcenter{\img[][50pt][49pt][{$\xymatrix{ X \ar[r]^-f \ar[d] & E \ar[d]^p\\ Y \ar[r]^-F & B.}$}]{Images/imgfa14f7076984aa23c4481aa20fbb3894.svg}} \end{aligned} {\tag{2.1}} \end{equation}$$

If there exists a map $G : Y \to E$ lifting $F$ and so that $G\vert _X$ is fibrewise homotopic to $f$, then there is also a lift $H$ of $F$ so that $H\vert _X=f$. To see this, choose a fibrewise homotopy $\varphi : [0,1] \times X \to E$ from $G\vert _X$ to $f$, let $J = ([0,1] \times X) \cup (\{0\} \times Y) \subset [0,1] \times Y$, and write $\varphi \cup G : J \to E$ for the map induced by $\varphi$ and $G$. The following diagram is then commutative

$$\begin{equation*} \vcenter{\img[][118pt][52pt][{$\xymatrix{ J \ar[rr]^-{\varphi\cup G} \ar[d] & & E \ar[d]^p\\ [0,1] \times Y \ar[r]^-{\pi_Y} & Y \ar[r]^-{F} & B, }$}]{Images/img5972085807c8f6541ea351d9e9541df2.svg}} \end{equation*}$$

and by the microfibration property there is a lift $g : U \to E$ defined on an open neighbourhood $U$ of $J$. Let $\phi : Y \to [0,1]$ be a continuous function with graph inside $U$ and so that $X \subset \phi ^{-1}(1)$. Then we set $H(y) = g(\phi (y),y)$; this is a lift of $F$ as $g$ is a lift of $F \circ \pi _Y$, and if $y \in X$, then $\phi (y)=1$ and so $H(y) = g(1,y) = f(y)$, as required.

Weiss proved in Wei05, Lemma 2.2 that if $f: E \to B$ is a Serre microfibration with weakly contractible fibres (i.e. $f^{-1}(b)$ is weakly contractible for all $b \in B$), then $f$ is in fact a Serre fibration and hence a weak homotopy equivalence. We shall need the following generalisation, whose proof is essentially the same as that of Weiss.

Proposition 2.6.

Let $p: E \to B$ be a Serre microfibration such that $p^{-1}(b)$ is $n$-connected for all $b \in B$. Then the homotopy fibres of $p$ are also $n$-connected; i.e. the map $p$ is $(n+1)$-connected.

Proof.

Let us first prove that $p^I: E^I \to B^I$ is a Serre microfibration with $(n-1)$-connected fibres, where $X^I = \mathrm{Map}([0,1],X)$ is the space of (unbased) paths in $X$, equipped with the compact-open topology. Using the mapping space adjunction, it is obvious that $p^I$ is a Serre microfibration, and showing the connectivity of its fibres amounts to proving that any diagram of the form

$$\begin{equation*} \vcenter{\img[][140pt][52pt][{$\xymatrix{ [0,1] \times\partial D^k \ar[rr] \ar[d] & & E\ar[d]^p\\ [0,1] \times D^k \ar[r]^-{\text{proj}}& [0,1] \ar[r] & B }$}]{Images/imgb8a6f6bff9c86ebb65ac118974aa70f9.svg}} \end{equation*}$$

with $k \leq n$ admits a diagonal $h: [0,1] \times D^k \to E$. Since fibres of $p$ are $(k-1)$-connected (in fact $k$-connected), such a diagonal can be found on each $\{a\} \times D^k$, and by the microfibration property these lifts extend to a neighbourhood. By the Lebesgue number lemma we may therefore find an integer $N \gg 0$ and lifts $h_i: [(i-1)/N,i/N]\times D^k \to E$ for $i = 1, \dots , N$. The two restrictions $h_i, h_{i+1}: \{i/N\} \times D^k \to E$ agree on $\{i/N\} \times \partial D^k$ and map into the same fibre of $p$. Since these fibres are $k$-connected, the restrictions of $h_i$ and $h_{i+1}$ are homotopic relative to $\{i/N\} \times \partial D^k$ as maps into the fibre, and we may use diagram 2.1 with $Y = [i/N,(i+1)/N] \times D^k$ and $X = (\{i/N\}\times D^k) \cup ([i/N,(i+1)/N] \times \partial D^k)$ to inductively replace $h_{i+1}$ by a homotopy which can be concatenated with $h_i$. The concatenation of the $h_i$s then gives the required diagonal.

Let us now prove that for all $k \leq n$, any lifting diagram

$$\begin{align*} \begin{aligned} \vcenter{\img[][81pt][55pt][{$\xymatrix{ \{0\} \times I^k \ar[r]^-f \ar[d] & E \ar[d]^p\\ [0,1] \times I^k \ar@{..>}[ur]^H\ar[r]^-h & B }$}]{Images/imge530c7550ee6b79d5f07b65b2c050537.svg}} \end{aligned} \end{align*}$$

admits a diagonal map $H$ making the diagram commutative. To see this, we first use that fibres of the map $p^{I^{k+1}}: E^{I^{k+1}} \to B^{I^{k+1}}$ are non-empty (in fact $(n-k-1)$-connected) to find a diagonal $G$ making the lower triangle commute. The restriction of $G$ to $\{0\} \times I^k$ need not agree with $f$, but they lie in the same fibre of $p^{I^k}: E^{I^k} \to B^{I^k}$. Since this map has path-connected fibres, these are fibrewise homotopic, and hence we may apply 2.1 to replace $G$ with a lift $H$ making both triangles commute.

By a standard argument (see Spa66, page 375) the above homotopy lifting property implies that $p: E \to B$ has the homotopy lifting property with respect to all CW-complexes of dimension $\leq n$. By another standard argument Spa66, page 376 this implies that for any $e \in E$ with $b=p(e)$ the map $p_*: \pi _i(E, p^{-1}(b), e) \to \pi _i(B, \{b\}, b)$ is surjective for $i \leq n+1$ and injective for $i \leq n$. As $p^{-1}(b)$ is $n$-connected, the long exact sequence for the pair $(E, p^{-1}(b))$ shows that $\pi _i(E, e) \to \pi _i(E, p^{-1}(b), e)$ is surjective for $i \leq n+1$ and injective for $i \leq n$, so it follows that $p_*: \pi _i(E, e) \to \pi _i(B, b)$ is surjective for $i \leq n+1$ and injective for $i \leq n$ for all basepoints; i.e. $p$ is $(n+1)$-connected.

■

2.3. Semisimplicial sets and spaces

Let $\Delta _\mathrm{inj}^*$ be the category whose objects are the ordered sets $[p] = (0 < \cdots < p)$ with $p \geq -1$, and whose morphisms are the injective order preserving functions. An augmented semisimplicial set is a contravariant functor $X$ from $\Delta _\mathrm{inj}^*$ to the category of sets. As usual, such a functor is specified by the sets $X_p = X([p])$ and face maps $d_i: X_p \to X_{p-1}$ for $i = 0, \dots , p$. A (non-augmented) semisimplicial set is a functor defined on the full subcategory $\Delta _\mathrm{inj}$ on the objects with $p \geq 0$ and is the same thing as is sometimes called a “$\Delta$-complex”. Semisimplicial spaces are defined similarly.

Let us briefly discuss the relationship between simplicial complexes and semisimplicial sets. To any simplicial complex $K$ there is an associated semisimplicial set $K_\bullet$, whose $p$-simplices are the injective simplicial maps $\Delta ^p \to K$, i.e. ordered $(p+1)$-tuples of distinct vertices in $K$ spanning a $p$-simplex. There is a natural surjection $|K_\bullet | \to |K|$, and any choice of total order on the set of vertices of $K$ induces a splitting $|K| \to |K_\bullet |$. In particular, $|K|$ is at least as connected as $|K_\bullet |$.

We shall use the following well known result.

Proposition 2.7.

Let $f_\bullet : X_\bullet \to Y_\bullet$ be a map of semisimplicial spaces such that $f_p: X_p \to Y_p$ is $(n-p)$-connected for all $p$. Then $|f_\bullet |: |X_\bullet | \to |Y_\bullet |$ is $n$-connected.

Proof.

We shall use the skeletal filtration of geometric realisations, and recall that $|X_\bullet |^{(q)}$ is homeomorphic to a pushout $|X_\bullet |^{(q-1)} \leftarrow X_q \times \partial \Delta ^q \to X_q \times \Delta ^q$ and similarly for $|Y_\bullet |$. By induction on $q$ we will prove that $|X_\bullet |^{(q)} \to |Y_\bullet |^{(q)}$ is $n$-connected for all $q$. The case $q = -1$ is vacuous, so we proceed to the induction step. As $f_q : X_q \to Y_q$ is $(n-q)$-connected, we can find a factorisation

$$X_q \overset{g_q}{\longrightarrow }Z_q \overset{h_q}{\longrightarrow }Y_q,$$

where $h_q$ is a weak homotopy equivalence, and the map $g_q$ is a relative CW-complex which only has cells of dimensions strictly greater than $n-q$. Define a new semisimplicial space $Z_\bullet$ with $Z_q$ in degree $q$ and

$$Z_i = \begin{cases} Y_i & \text{ if $i < q$}\\ X_i & \text{ if $i > q$} \end{cases}$$

and the evident face maps (in particular $d_i^Z : Z_q \to Z_{q-1}=Y_{q-1}$ is $d_i^Y \circ h_q$). There is then a factorisation $f_\bullet = h_\bullet \circ g_\bullet : X_\bullet \to Z_\bullet \to Y_\bullet$, and the map $|h_\bullet |^{(q)}: |Z_\bullet |^{(q)} \to |Y_\bullet |^{(q)}$ is a weak homotopy equivalence because $h_i$ is a weak homotopy equivalence for all $i \leq q$.

By induction, the map $|X_\bullet |^{(q-1)} \to |Z_\bullet |^{(q-1)} = |Y_\bullet |^{(q-1)}$ is $n$-connected, and hence the map from $|X_\bullet |^{(q)}$ to the pushout of $|Z_\bullet |^{(q-1)} \leftarrow X_q \times \partial \Delta ^q \to X_q \times \Delta ^q$ is also $n$-connected. Since $|Z_\bullet |^{(q)}$ is obtained from this pushout by attaching cells of dimension strictly greater than $n$, namely $\Delta ^q$ times the relative cells of $(Z_q,X_q)$, the map $|g_\bullet |^{(q)}$ is the composition of an $n$-connected map and a relative CW-complex with cells of dimension strictly greater than $n$, and hence $n$-connected.

■

The following is the main result of this section.

Proposition 2.8.

Let $Y_\bullet$ be a semisimplicial set and $Z$ be a Hausdorff space. Let $X_\bullet \subset Y_\bullet \times Z$ be a sub-semisimplicial space which in each degree is an open subset. Then $\pi : |X_\bullet | \to Z$ is a Serre microfibration.

Proof.

Let us write $p : |X_\bullet | \to |Y_\bullet \times Z| \to |Y_\bullet |$ for the map induced on geometric realisations by the inclusion and projection. For $\sigma \in Y_n$, let us write $Z_\sigma \subset Z$ for the open subset defined by $(\{\sigma \} \times Z) \cap X_n = \{\sigma \} \times Z_\sigma$. Points in $|X_\bullet |$ are described by data

$$(\sigma \in Y_n ;\, z \in Z_\sigma ;\, (t_0, \ldots , t_n) \in \Delta ^n)$$

up to the evident relation when some $t_i$ is zero, but we emphasise that the continuous, injective map $\iota = p \times \pi : |X_\bullet | \hookrightarrow |Y_\bullet | \times Z$ will not generally be a homeomorphism onto its image, as the quotient of a subspace is not always a subspace of the quotient. (For example, let $Y_0 = \{0,1\}$, $Y_1 = \{\iota \}$, $d_0 \iota = 0$, $d_1 \iota = 1$, and $Y_i = \emptyset$ for $i > 1$, $Z=[0,1]$, $X_0 = (\{0\} \times (0,1]) \cup (\{1\} \times [0,1]),$ and $X_1 = \{\iota \} \times (0,1]$. Then the image of $|X_\bullet | \to |Y_\bullet | \times Z \cong [0,1] \times [0,1]$ is $([0,1] \times (0,1]) \cup (\{1\} \times [0,1])$ but the inverse map is not continuous at $(1,0)$. In fact $|X_\bullet |$ is not first countable, so it is not homeomorphic to any subspace of $[0,1] \times [0,1]$.) This is not a problem: in fact we shall make use of the topology on $|X_\bullet |$ being finer than the subspace topology.

Suppose now given a lifting problem

$$\begin{equation*} \vcenter{\img[][90pt][56pt][{$\xymatrix{ \{0\} \times D^k \ar[r]^-{f} \ar[d] & \vert X_\bullet\vert\ar[d]^{\pi}\\ [0,1] \times D^k \ar[r]^-{F} & Z. }$}]{Images/img786bdac70a1d3ac97b87d77de9373d67.svg}} \end{equation*}$$

The composition $D^k \overset{f}{\to }|X_\bullet | \overset{p}{\to }|Y_\bullet |$ is continuous, so the image of $D^k$ is compact and hence contained in a finite subcomplex, and it intersects finitely many open simplices $\{\sigma _i\} \times \mathrm{int}(\Delta ^{n_i})\subset |Y_\bullet |$. The sets $C_{\sigma _i} = (p \circ f)^{-1}(\{\sigma _i\} \times \mathrm{int}(\Delta ^{n_i}))$ then cover $D^k$, and their closures $\overline{C}_{\sigma _i}$ give a finite cover of $D^k$ by closed sets. Let us write $f\vert _{{C}_{\sigma _i}}(x) = (\sigma _i;z(x);t(x))$, with $z(x) \in Z_{\sigma _i} \subset Z$ and $t(x) = (t_0(x), \ldots , t_{n_i}(x)) \in \mathrm{int}(\Delta ^{n_i})$.

Certainly $\pi \circ f$ sends the set $C_{\sigma _i}$ into the open set $Z_{\sigma _i}$, but in fact the following stronger property is true.

Claim. The map $\pi \circ f$ sends the set $\overline{C}_{\sigma _i}$ into $Z_{\sigma _i}$.

Proof of claim.

We consider a sequence $(x^j) \in C_{\sigma _i}$, $j \in \mathbb{N}$, converging to a point $x \in \overline{C}_{\sigma _i} \subset D^k$, and we shall verify that $z= \pi \circ f(x) \in Z_{\sigma _i}$.

As $f$ is continuous, the sequence $f(x^j) = (\sigma _i;z(x^j);t(x^j)) \in |X_\bullet |$ converges to $f(x)$, and, passing to a subsequence, we may assume that the $t(x^j)$ converge to a point $t\in \Delta ^{n_i}$. The subset

$$\begin{equation*} A = \{f(x^j)\,\, \vert \,\, j \in \mathbb{N}\} \subset |X_\bullet | \end{equation*}$$

is contained in $\pi ^{-1}(Z_{\sigma _i})$ and has $f(x)$ as a limit point in $|X_\bullet |$, so if $z = \pi (f(x)) \not \in Z_{\sigma _i}$, then the set $A$ is not closed in $|X_\bullet |$.

For a contradiction we will show that $A$ is closed, by proving that its inverse image in $\coprod _\tau \{\tau \} \times Z_\tau \times \Delta ^{|\tau |}$ is closed, where the coproduct is over all simplices $\tau \in \coprod _n Y_n$. The inverse image in $\{\sigma _i\} \times Z_{\sigma _i} \times \Delta ^{n_i}$ is

$$\begin{equation*} B = \{(\sigma _i;\, z(x^j);\, t(x^j))\,\, \vert \,\, j \in \mathbb{N}\}, \end{equation*}$$

which is closed (since $Z$ is Hausdorff, taking the closure in $\{\sigma _i\} \times Z \times \Delta ^{n_i}$ adjoins only the point $(\sigma _i; z; t)$, which by assumption is outside $\{\sigma _i\} \times Z_{\sigma _i} \times \Delta ^{n_i}$). If $\sigma _i = \theta ^*(\tau )$ for a morphism $\theta \in \Delta _{\mathrm{inj}}$, we have $Z_\tau \subset Z_{\sigma _i}$ and hence $B \cap (\{\sigma _i\} \times Z_\tau \times \Delta ^{|\sigma _i|})$ is closed in $\{\sigma _i\} \times Z_\tau \times \Delta ^{|\sigma _i|}$, so applying $\theta _*: \Delta ^{|\sigma _i|} \to \Delta ^{|\tau |}$ gives a closed subset $B_\theta \subset \{\tau \} \times Z_\tau \times \Delta ^{|\tau |}$. The inverse image of $A$ in $\{\tau \} \times Z_\tau \times \Delta ^{|\tau |}$ is the union of the $B_\theta$ over the finitely many $\theta$ with $\theta ^*(\tau ) = \sigma _i$ and is hence closed.

■

We have a continuous map $F_i = F\vert _{[0,1] \times \overline{C}_{\sigma _i}} : [0,1] \times \overline{C}_{\sigma _i} \to Z$, and by the claim $F_i^{-1}(Z_{\sigma _i})$ is an open neighbourhood of the compact set $\{0\} \times \overline{C}_{\sigma _i}$, so there is an $\varepsilon _i > 0$ such that $F_i([0,\varepsilon _i] \times \overline{C}_{\sigma _i}) \subset Z_{\sigma _i}$. We set $\varepsilon = \min _i (\varepsilon _i)$ and define the lift

$$\widetilde{F}_i(s, x) = (\sigma _i;F_i(s,x);t(x)) : [0,\varepsilon ] \times \overline{C}_{\sigma _i} \longrightarrow \{\sigma _i\} \times Z_{\sigma _i} \times \Delta ^{n_i} \longrightarrow |X_\bullet |,$$

which is clearly continuous. The functions $\widetilde{F}_i$ and $\widetilde{F}_j$ agree where they are both defined, and so these glue to give a continuous lift $\widetilde{F}$ as required.

■

Corollary 2.9.

Let $Z$, $Y_\bullet$, and $X_\bullet$ be as in Proposition 2.8. For $z \in Z$, let $X_\bullet (z) \subset Y_\bullet$ be the sub-semisimplicial set defined by $X_\bullet \cap (Y_\bullet \times \{z\}) = X_\bullet (z) \times \{z\}$ and suppose that $|X_\bullet (z)|$ is $n$-connected for all $z \in Z$. Then the map $\pi : |X_\bullet | \to Z$ is $(n+1)$-connected.

Proof.

This follows by combining Propositions 2.6 and 2.8, once we prove that $|X_\bullet (z)|$ is homeomorphic to $\pi ^{-1}(z)$ (in the subspace topology from $|X_\bullet |$). Since $X_\bullet (z)\subset Y_\bullet$, the composition $|X_\bullet (z)| \to |X_\bullet | \to |Y_\bullet |$ is a homeomorphism onto its image. It follows that $|X_\bullet (z)| \to |X_\bullet |$ is a homeomorphism onto its image, which is easily seen to be $\pi ^{-1}(z)$.

■

3. Algebra

We fix $\varepsilon = \pm 1$ and let $\Lambda \subset \mathbb{Z}$ be a subgroup satisfying

$$\{a - \varepsilon {a} \,\,\vert \,\, a \in \mathbb{Z}\} \subset \Lambda \subset \{a \in \mathbb{Z}\,\,\vert \,\, a + \varepsilon {a} =0\}.$$

Following Bak Bak69Bak81, we call such a pair $(\varepsilon , \Lambda )$ a form parameter. Since we work over the ground ring $\mathbb{Z}$, there are only three options for $(\varepsilon ,\Lambda )$, namely $(+1, \{0\})$, $(-1,2\mathbb{Z}),$ and $(-1,\mathbb{Z})$. An $(\varepsilon ,\Lambda )$-quadratic module is a triple $\mathsf{M}= (M, \lambda , \mu )$ where $M$ is a $\mathbb{Z}$-module, $\lambda : M \otimes M \to \mathbb{Z}$ is bilinear and satisfies $\lambda (x,y) = \varepsilon \lambda (y,x)$, and $\mu : M \to \mathbb{Z}/\Lambda$ is a quadratic form whose associated bilinear form is $\lambda$ reduced modulo $\Lambda$. By this we mean that $\mu$ is a function such that

(i)

$\mu (a \cdot x) = a^2 \cdot \mu (x)$ for $a \in \mathbb{Z}$,

(ii)

$\mu (x+y) - \mu (x) - \mu (y) \equiv \lambda (x, y) \mod \Lambda$.

We say the quadratic module $\mathsf{M}= (M,\lambda ,\mu )$ is non-degenerate if the map

$$\begin{align*} \lambda ^\vee : M &\longrightarrow M^\vee \\ x &\longmapsto \lambda (-, x) \end{align*}$$

is an isomorphism. If $\mathsf{M}= (M,\lambda _{\mathsf{M}},\mu _{\mathsf{M}})$ and $\mathsf{N}= (N,\lambda _{\mathsf{N}},\mu _{\mathsf{N}})$ are quadratic modules with the same form parameter, then their (orthogonal) direct sum is $\mathsf{M}\oplus \mathsf{N}= (M \oplus N,\lambda _\oplus ,\mu _\oplus )$ where $\lambda _\oplus (m + n,m'+n') = \lambda _{\mathsf{M}}(m,m') + \lambda _{\mathsf{N}}(n+n')$ and $\mu _\oplus (m+n) = \mu _{\mathsf{M}}(m) + \mu _{\mathsf{N}}(n)$. A morphism $f: \mathsf{M}\to \mathsf{N}$ is a homomorphism $f: M \to N$ with the properties that $\lambda _{\mathsf{N}} \circ (f \otimes f) = \lambda _{\mathsf{M}}$ and $\mu _{\mathsf{M}} = \mu _{\mathsf{N}} \circ f$. If $\mathsf{M}$ is non-degenerate, any such morphism is split injective because

$$\begin{equation*} M \overset{f}{\longrightarrow }N \overset{\lambda _N^\vee }{\longrightarrow }N^\vee \overset{f^*}{\longrightarrow }M^\vee \end{equation*}$$

is an isomorphism, and in fact induces an isomorphism $\mathsf{N} \cong \mathsf{M} \oplus f(\mathsf{M})^\perp$ of $(\varepsilon ,\Lambda )$-quadratic modules.

The hyperbolic module $\mathsf{H}$ is the non-degenerate $(\varepsilon ,\Lambda )$-quadratic module with underlying abelian group free of rank 2 with basis $e$ and $f$, and the unique quadratic module structure with $\lambda _\mathsf{H}(e,f) = 1$, $\lambda _\mathsf{H}(e,e) = \lambda _\mathsf{H}(f,f) = 0$, and $\mu _\mathsf{H}(e) = \mu _\mathsf{H}(f) = 0$. We write $\mathsf{H}^g$ for the orthogonal direct sum of $g$ copies of $\mathsf{H}$ and define the Witt index of an $(\varepsilon , \Lambda )$-quadratic module $\mathsf{M}$ as

$$\begin{equation*} g(\mathsf{M}) = \sup \{g \in \mathbb{N}\,\,\vert \,\, \text{there exists a morphism $\mathsf{H}^g \to \mathsf{M}$} \}. \end{equation*}$$

The Witt index obviously satisfies $g(\mathsf{M}\oplus \mathsf{H}) \geq g(\mathsf{M}) + 1$, and we shall also consider the stable Witt index defined by

$$\begin{equation} \overline{g}(\mathsf{M}) = \sup \{g(\mathsf{M}\oplus \mathsf{H}^{k}) - k \mid k \geq 0\}. {\tag{3.1}} \end{equation}$$

This satisfies $\overline{g}(\mathsf{M}\oplus \mathsf{H}) = \overline{g}(\mathsf{M}) +1$ for all $\mathsf{M}$.

Definition 3.1.

For a quadratic module $\mathsf{M}$, let $K^a(\mathsf{M})$ be the simplicial complex whose vertices are morphisms $h:\mathsf{H}\to \mathsf{M}$ of quadratic modules. The set $\{h_0, \ldots , h_p\}$ forms a $p$-simplex if the submodules $h_i(\mathsf{H}) \subset \mathsf{M}$ are orthogonal with respect to $\lambda$ (we impose no additional condition on the quadratic forms).

The complex $K^a(\mathsf{M})$ is almost the same as one considered by Charney Cha87, which she proves to be highly connected when $\mathsf{M}= \mathsf{H}^{g}$. We shall need a connectivity theorem for more general $\mathsf{M}$, assuming only that $\overline{g}(\mathsf{M}) \geq g$. In particular, we do not wish to assume $\mathsf{M}$ is non-degenerate (or even that the underlying $\mathbb{Z}$-module is free). In Section 4 we shall give a self-contained proof of the following generalisation of Charney’s result.

Theorem 3.2.

Let $g \in \mathbb{N}$, and let $\mathsf{M}$ be a quadratic module with $\overline{g}(\mathsf{M}) \geq g$. Then the geometric realisation $|K^a(\mathsf{M})|$ is $\lfloor \tfrac{g-4}{2} \rfloor$-connected, and $\mathrm{lCM}(K^a(\mathsf{M})) \geq \lfloor \tfrac{g-1}{2} \rfloor$.

Before embarking on the proof, let us deduce two consequences of the path-connectedness of $|K^a(\mathsf{M})|$.

Proposition 3.3 (Transitivity).

If $|K^a(\mathsf{M})|$ is path-connected and $h_0, h_1 : \mathsf{H}\to \mathsf{M}$ are morphisms of quadratic modules, then there is an isomorphism of quadratic modules $f : \mathsf{M}\to \mathsf{M}$ such that $h_1 = f \circ h_0$.

Proof.

Suppose first that $h_0$ and $h_1$ are orthogonal. Then there is an orthogonal decomposition

$$\begin{equation*} \mathsf{M}\cong h_0(\mathsf{H}) \oplus h_1(\mathsf{H}) \oplus \mathsf{M}' \end{equation*}$$

and so an evident automorphism of quadratic modules which swaps the $h_i(\mathsf{H})$. Now, the relation between morphisms $e : \mathsf{H}\to \mathsf{M}$ of differing by an automorphism is an equivalence relation, and we have just shown that adjacent vertices in $K^a(\mathsf{M})$ are equivalent. If the complex is path-connected, then all vertices are equivalent.

■

Proposition 3.4 (Cancellation).

Suppose that $\mathsf{M}$ and $\mathsf{N}$ are quadratic modules and there is an isomorphism $\mathsf{M}\oplus \mathsf{H}\cong \mathsf{N}\oplus \mathsf{H}$. If $|K^a(\mathsf{M}\oplus \mathsf{H})|$ is path-connected, then there is also an isomorphism $\mathsf{M}\cong \mathsf{N}$.

Proof.

An isomorphism $\varphi : \mathsf{M}\oplus \mathsf{H}\to \mathsf{N}\oplus \mathsf{H}$ gives a morphism $\varphi \vert _\mathsf{H}: \mathsf{H}\to \mathsf{N}\oplus \mathsf{H}$ of quadratic modules, and we also have the standard inclusion $\iota : \mathsf{H}\to \mathsf{N}\oplus \mathsf{H}$. By Proposition 3.3, these differ by an automorphism of $\mathsf{N}\oplus \mathsf{H}$, so in particular their orthogonal complements are isomorphic.

■

By Theorem 3.2, the complex $K^a(\mathsf{M})$ is path-connected provided $\overline{g}(\mathsf{M}) \geq 4$. As long as $\overline{g}(\mathsf{M}) \geq 3$, Proposition 3.4 therefore gives the implication $(\mathsf{M}\oplus \mathsf{H}\cong \mathsf{N}\oplus \mathsf{H}) \Rightarrow (\mathsf{M}\cong \mathsf{N})$. It follows that as long as $\overline{g}(\mathsf{M}) \geq 3$ we have $g(\mathsf{M}\oplus \mathsf{H}) = g(\mathsf{M}) + 1$ and hence $g(\mathsf{M}) = \overline{g}(\mathsf{M})$, but for the inductive proof of Theorem 3.2 it is more convenient to work with $\overline{g}$.

4. Proof of Theorem 3.2

Proposition 4.1.

Let $\mathrm{Aut}(\mathsf{H}^{n+1})$ act on $\mathsf{H}^{n+1}$, and consider the orbits of elements of $\mathsf{H}\oplus 0 \subset \mathsf{H}^{n+1}$. Then we have

$$\begin{align*} \mathrm{Aut}(\mathsf{H}^{n+1}) \cdot (\mathsf{H}\oplus 0) = \mathsf{H}^{n+1}. \end{align*}$$

Proof.

We consider the form parameters $(+1,\{0\})$ and $(-1,2\mathbb{Z})$ separately. The case $(-1,\mathbb{Z})$ follows from the case $(-1,2\mathbb{Z})$, as the automorphism group is larger. Recall that a vector $v = (a_0,b_0, \dots , a_n,b_n) \in \mathsf{H}^{n+1}$ is called unimodular if its coordinates have no common divisor. Any $v \in \mathsf{H}^{n+1}$ can be written as $v = d v'$ with $d \in \mathbb{N}$ and $v' \in \mathsf{H}^{n+1}$ unimodular, so it suffices to prove that for any unimodular $v \in \mathsf{H}^{n+1}$ there exists $\phi \in \mathrm{Aut}(\mathsf{H}^{n+1})$ with $\phi (v) \in \mathsf{H}\oplus 0$.

In the case of form parameter $(+1,\{0\})$, this follows from Wal62, Theorem 1, which asserts that $\mathrm{Aut}(\mathsf{H}^{n+1})$ acts transitively on unimodular vectors of a given length. Therefore, any unimodular vector $v \in \mathsf{H}^{n+1}$ is in the same orbit as $(1,a,0,\dots , 0) \in \mathsf{H}\oplus 0$ for $a = (\lambda (v,v))/2 = \mu (v)\in \mathbb{Z}$.

The case $(-1,2\mathbb{Z})$ can be proved in a manner similar to Wal62, Theorem 1. First, in $\mathrm{Aut}(\mathsf{H})$ we have the transformations $(a,b) \mapsto \pm (b,-a)$, so any orbit has a representative with $0 \leq b \leq |a|$. For $0 < b < a$, the transformation $(a,b) \mapsto (a - 2b,b)$ will decrease the number $\max (|a|,|b|)$, and for $0 < b < -a$, the inverse transformation will do the same; so inductively we see that any orbit has a representative of the form $(a,0)$ or $(a,a)$ for some integer $a \geq 0$. It follows that under $\mathrm{Aut}(\mathsf{H}) \times \mathrm{Aut}(\mathsf{H}) \leq \mathrm{Aut}(\mathsf{H}^2)$ acting on $\mathsf{H}^2$, the orbit of a unimodular vector has a representative of the form $v = (a,b,c,d)$ with $b\in \{0,a\}$, $d \in \{0,c\}$, and $\gcd (a,c)=1$. On such a representative we then use the transformation

$$\begin{equation*} (a,b,c,d) \longmapsto (a,b+c,c,d+a), \end{equation*}$$

and since $\gcd (a,b+c)=1 = \gcd (c,d+a)$, we can use $\mathrm{Aut}(\mathsf{H}) \times \mathrm{Aut}(\mathsf{H})$ to get to a representative with $a=c=1$ and $b,d \in \{0,1\}$. We can act on this representative $(1,b,1,d)$ by the element of $\mathrm{Aut}(\mathsf{H}^2)$ given by right multiplication by the matrix

$$\begin{pmatrix} 1 & d & -d & 1 \\ 0 & 1 & 0 & 0 \\ 0 & -d & 0 & -1 \\ 0 & 1 & 1 & 0 \end{pmatrix}$$

to get the representative $(1,b+d,0,0) \in \mathsf{H}\oplus 0 \leq \mathsf{H}^2$. This proves the case $n=1$, and the general case follows from this by induction.

■

Corollary 4.2.

Let $\mathsf{M}$ be a quadratic module with $g(\mathsf{M}) \geq g$, and let $\ell : M \to \mathbb{Z}$ be linear. Then the quadratic form $\mathsf{K} = (\operatorname *{Ker}(\ell ), \lambda \vert _{\operatorname *{Ker}(\ell )}, \mu \vert _{\operatorname *{Ker}(\ell )})$ satisfies $g(\mathsf{K}) \geq g-1$. Similarly if $\overline{g}(\mathsf{M}) \geq g$, then $\overline{g}(\mathsf{K}) \geq g-1$.

Proof.

We can find a morphism $\phi : \mathsf{H}^g \to \mathsf{M}$. By non-degeneracy of the form on $\mathsf{H}^g$, the composite $\ell \circ \phi \in \mathrm{Hom}(\mathsf{H}^g,\mathbb{Z})$ is of the form $\lambda (x,-)$. By Proposition 4.1 we can, after precomposing $\phi$ with an automorphism, assume $x \in \mathsf{H}\oplus 0 \subset \mathsf{H}\oplus \mathsf{H}^{g-1}$, and hence $0 \oplus \mathsf{H}^{g-1} \subset \operatorname *{Ker}(\ell )$, so $\phi$ restricts to a morphism $\mathsf{H}^{g-1} \to \mathsf{K}$. The claim about the stable Witt index follows from the unstable by considering $\mathsf{M}\oplus \mathsf{H}^k$.

■

We may deduce the first non-trivial cases of Theorem 3.2 from this corollary.

Proposition 4.3.

If $\overline{g}(\mathsf{M}) \geq 2$, then $K^a(\mathsf{M}) \neq \emptyset$, and if $\overline{g}(\mathsf{M}) \geq 4$, then $K^a(\mathsf{M})$ is path-connected.

Proof.

We consider the second case first. Let us first make the stronger assumption that the (unstable) Witt index is $g(\mathsf{M}) \geq 4$. Then there exists an $h_0: \mathsf{H}\to \mathsf{M}$ with $g(h_0(\mathsf{H})^\perp ) \geq 3$. Any $h: \mathsf{H}\to \mathsf{M}$ then gives rise to a map of $\mathbb{Z}$-modules,

$$\begin{equation*} h_0(\mathsf{H})^\perp \longrightarrow \mathsf{M}\longrightarrow h(\mathsf{H}), \end{equation*}$$

where the first map is the inclusion and the second is orthogonal projection. The kernel of this map is $h_0(\mathsf{H})^\perp \cap h(\mathsf{H})^\perp$. This is the intersection of the kernels of two linear maps on $h_0(\mathsf{H})^\perp$, so by Corollary 4.2 we have $g(h_0(\mathsf{H})^\perp \cap h(\mathsf{H})^\perp ) \geq 1$ and we can find an $h_1 \in K^a(h_0(\mathsf{H})^\perp \cap h(\mathsf{H})^\perp )$. Then $\{h_0,h_1\}$ and $\{h_1,h\}$ are 1-simplices in $K^a(\mathsf{M})$, so there is a path (of length at most 2) from any vertex to $h_0$.

The general case $\overline{g}(\mathsf{M}) \geq 4$ can be reduced to this by an argument as in the last paragraph of Section 3. Indeed, we can write $\mathsf{M}\oplus \mathsf{H}^k \cong \mathsf{N}\oplus \mathsf{H}^k$ for some integer $k \geq 0$ and quadratic module $\mathsf{N}$ with $g(\mathsf{N}) \geq 4$ and use the connectivity of $K^a(\mathsf{N}\oplus \mathsf{H}^j)$ for all $j \geq 0$ to inductively deduce from Proposition 3.4 that $\mathsf{M}\cong \mathsf{N}$ and in particular $g(\mathsf{M}) = g(\mathsf{N}) \geq 4$.

Similarly, if $\overline{g}(\mathsf{M}) \geq 2$ we can write $\mathsf{M}\oplus \mathsf{H}^k \cong \mathsf{N}\oplus \mathsf{H}^{k}$ with $g(\mathsf{N}) \geq 2$, and inductively use Proposition 3.4 to see $\mathsf{M}\oplus \mathsf{H}\cong \mathsf{N}\oplus \mathsf{H}$. As in the first part of the proof, $\mathsf{M}$ is isomorphic to the intersection of the kernels of two linear maps $\mathsf{N}\oplus \mathsf{H}\to \mathbb{Z}$. By Corollary 4.2 the Witt index drops by at most one for each linear map, so the Witt index of $\mathsf{M}$ is at least 1.

■

Proof of Theorem 3.2.

We proceed by induction on $g$. At each stage of the induction, the statement $\mathrm{lCM}(K^a(\mathsf{M})) \geq \lfloor \tfrac{g-1}{2}\rfloor$ follows easily from the induction hypothesis. Indeed, a $p$-simplex $\sigma = \{h_0, \ldots , h_p\} < K^a(\mathsf{M})$ induces (after choosing an ordering of its vertices) a canonical splitting $\mathsf{M}\cong \mathsf{M}' \oplus \mathsf{H}^{p+1}$, where $\mathsf{M}' = (\bigoplus h_i (\mathsf{H}))^\perp \subset \mathsf{M}$. We then have an isomorphism of simplicial complexes $\mathrm{Lk}(\sigma ) \cong K^a(\mathsf{M}')$, and we have $\overline{g}(\mathsf{M}') = \overline{g}(\mathsf{M}) - p-1$. By induction, $\mathrm{Lk}(\sigma )$ is then $\lfloor \tfrac{(g-p-1)-4}{2} \rfloor$-connected, and by the inequality

$$\begin{equation*} \big \lfloor \tfrac{g-p-1-4}{2}\big \rfloor \geq \big \lfloor \tfrac{g-5}{2}\big \rfloor - p = \big \lfloor \tfrac{g-1}{2}\big \rfloor -p -2 \end{equation*}$$

we see that the link of any $p$-simplex is $(\lfloor \tfrac{g-1}{2}\rfloor -p -2)$-connected as required.

It remains to prove the statement about connectivity of $K^a(\mathsf{M})$. The connectivity statement is void for $g\leq 1$. For $g=2,3$ we assert that $|K^a(\mathsf{M})| \neq \emptyset$ and for $g=4,5$ we assert that $|K^a(\mathsf{M})|$ is path-connected; both are covered by Proposition 4.3. For the induction step, let us assume that Theorem 3.2 holds up to $g-1$, and let $\mathsf{M}$ be a quadratic module with $\overline{g}(\mathsf{M}) \geq g$. By Proposition 4.3 we have $K^a(\mathsf{M}) \neq \emptyset$ so we may pick some $h: \mathsf{H}\to \mathsf{M}$. If we write $\mathsf{M}' = h(\mathsf{H})^\perp \subset \mathsf{M}$, we have $\mathsf{M}\cong \mathsf{M}' \oplus \mathsf{H}$ and $h(e)^\perp = \mathsf{M}' \oplus \mathbb{Z}e$. The inclusion $\mathsf{M}' \hookrightarrow \mathsf{M}$ may then be factored as $\mathsf{M}' \hookrightarrow \mathsf{M}' \oplus \mathbb{Z}e \hookrightarrow \mathsf{M}$, and we have an induced factorisation

$$\begin{equation} \vcenter{\img[][187pt][14pt][{$\xymatrix{ \vert K^a(\sM') \vert\ar@{^(->}[r]^-{\CircNum{1}} & \vert K^a(\sM' \oplus\bZ e) \vert\ar@{^(->}[r]^-{\CircNum{2}} & \vert K^a(\sM) \vert. }$}]{Images/imgc9ce4c707b9e051a96508b7405fd068b.svg}} {\tag{4.1}} \end{equation}$$

We now wish to show that Proposition 2.5 applies to the maps $①$ and $②$, both of which are inclusions of full subcomplexes, with $n = \lfloor \frac{g-4}{2} \rfloor$. For $①$, we use the projection $\pi : \mathsf{M}' \oplus \mathbb{Z}e \to \mathsf{M}'$. The summand $\mathbb{Z}e$ has the trivial quadratic structure and pairs trivially with anything in $\mathsf{M}' \oplus \mathbb{Z}e$, so $\pi$ is a morphism of quadratic modules and hence induces a retraction $\pi : K^a(\mathsf{M}' \oplus \mathbb{Z}e) \to K^a(\mathsf{M}')$. For any $p$-simplex $\sigma < K^a(\mathsf{M}' \oplus \mathbb{Z}e)$ we have

$$\begin{equation*} K^a(\mathsf{M}') \cap \mathrm{Lk}_{K^a(\mathsf{M}' \oplus \mathbb{Z}e)}(\sigma ) = \mathrm{Lk}_{K^a(\mathsf{M}')}(\pi (\sigma )), \end{equation*}$$

and to apply Proposition 2.5 we must show that this simplicial complex is $(n-p-1)$-connected. The splitting $\mathsf{M}\cong \mathsf{M}' \oplus \mathsf{H}$ shows that $\overline{g}(\mathsf{M}') \geq g-1$, so by induction we have $\mathrm{lCM}(K^a(\mathsf{M}')) \geq \lfloor \frac{g-2}{2}\rfloor$, and therefore the link of the $p$-simplex $\pi (\sigma )$ is $(\lfloor \frac{g-2}{2} \rfloor -p-2)$-connected. But $\lfloor \frac{g-2}{2} \rfloor -p-2 = n-p-1$.

For $②$, we first note that $\mathsf{M}' \oplus \mathbb{Z}e \subset \mathsf{M}$ is exactly the orthogonal complement $h(e)^\perp \subset \mathsf{M}$. For a $p$-simplex $\sigma = \{h_0, \dots , h_p\} \subset K^a(\mathsf{M})$ we write $\mathsf{M}'' = (\sum h_i(\mathsf{H}))^\perp \subset \mathsf{M}$ and have

$$\begin{equation*} K^a(\mathsf{M}' \oplus \mathbb{Z}e) \cap \mathrm{Lk}_{K^a(\mathsf{M})}(\sigma ) = K^a(\mathsf{M}'' \cap h(e)^\perp ). \end{equation*}$$

The isomorphism $\mathsf{M}\cong \mathsf{M}'' \oplus \mathsf{H}^{p+1}$ shows that $\overline{g}(\mathsf{M}'') \geq g-p-1$, and passing to the kernel of the linear functional $\lambda (h(e),-)\vert _{M''}$ reduces the stable Witt index by at most one by Corollary 4.2, so we have $\overline{g}(\mathsf{M}'' \cap h(e)^\perp ) \geq g-p-2$. By induction, the connectivity of $K^a(\mathsf{M}'' \cap h(e)^\perp )$ is therefore at least $\big \lfloor \tfrac{(g-p-2)-4}{2}\big \rfloor \geq n - p -1$.

We have shown that both inclusions $①$ and $②$ satisfy the hypothesis of Proposition 2.5, and therefore these maps are $n$-connected. The composition factors through the star of the vertex given by $h$ and is therefore nullhomotopic. This implies that $K^a(\mathsf{M})$ is $n$-connected, finishing the induction step.

■

5. Topology

Recall that in Section 1 we defined the manifold $W_{g,1} = D^{2n} \sharp g(S^n \times S^n)$ and that for a path-connected compact $2n$-manifold $W$ we defined $g(W)$ to be the maximal $g \in \mathbb{N}$ for which there exists an embedding $W_{g,1} \hookrightarrow W$. In analogy with 3.1 we define the stable genus of $W$ to be

$$\begin{equation*} \overline{g}(W) = \max \{g(W \sharp k(S^n \times S^n)) - k\mid k \in \mathbb{N}\}. \end{equation*}$$

Notice that $k \mapsto g(W \sharp k(S^n \times S^n)) - k$ is non-decreasing and bounded above by $b_n(W)/2$. In particular, the maximum is well-defined.

It will be convenient to have available the following small modification of the manifold $W_{1,1}$. First, we may choose an embedding $W_{1,1} \hookrightarrow S^n \times S^n$ with complement an open disc lying in $D^n_+ \times D^n_+$, the product of the two upper hemispheres, and from now on we shall implicitly use this embedding identify $W_{1,1}$ with a subset of $S^n \times S^n$. Let $H$ denote the manifold obtained from $W_{1,1}$ by gluing $[-1,0] \times D^{2n-1}$ to $\partial W_{1,1}$ along an orientation preserving embedding

$$\begin{equation*} \{-1\} \times D^{2n-1} \longrightarrow \partial W_{1,1}, \end{equation*}$$

which we also choose once and for all. This gluing of course does not change the diffeomorphism type (after smoothing corners), so $H$ is diffeomorphic to $W_{1,1}$, but contains a standard embedded $[-1,0] \times D^{2n-1} \subset H$. When we discuss embeddings of $H$ into a manifold with boundary $W$, we shall always insist that $\{0\} \times D^{2n-1}$ is sent into $\partial W$, and that the rest of $H$ is sent into the interior of $W$.

We shall also need a core $C \subset H$, defined as follows. Let $x_0 \in S^n$ be a basepoint. Let $S^n \vee S^n = (S^n \times \{x_0\}) \cup (\{x_0\} \times S^n) \subset S^n \times S^n$, which we may suppose is contained in $\mathrm{int}(W_{1,1})$. Choose an embedded path $\gamma$ in $H$ from $(x_0, -x_0)$ to $(0,0) \in [-1,0] \times D^{2n-1}$ whose interior does not intersect $S^n \vee S^n$, and whose image agrees with $[-1,0] \times \{0\}$ inside $[-1,0] \times D^{2n-1}$, and let

$$\begin{equation*} C = (S^n \vee S^n) \cup \mathrm{Im}(\gamma ) \cup (\{0\} \times D^{2n-1}) \subset H. \end{equation*}$$

The manifold $H$ is depicted together with $C \subset H$ in Figure 1.

Definition 5.1.

Let $W$ be a compact manifold, equipped with (the germ of) an embedding $c: (-\delta , 0] \times \mathbb{R}^{2n-1} \to W$ for some $\delta > 0$, such that $c^{-1}(\partial W) = \{0\} \times \mathbb{R}^{2n-1}$. Two embeddings $c$ and $c'$ define the same germ if they agree after making $\delta$ smaller.

(i)

Let $K_0(W) = K_0(W,c)$ be the space of pairs $(t,\phi )$, where $t \in \mathbb{R}$ and $\phi : H\to W$ is an embedding whose restriction to $(-1,0] \times D^{2n-1} \subset H$ satisfies that there exists an $\varepsilon \in (0,\delta )$ such that$$\begin{equation*} \phi (s,p) = c(s, p + te_1) \end{equation*}$$

for all $s \in (-\varepsilon ,0]$ and all $p \in D^{2n-1}$. Here, $e_1 \in \mathbb{R}^{2n-1}$ denotes the first basis vector.

(ii)

Let $K_p(W) \subset (K_0(W))^{p+1}$ consist of those tuples $((t_0, \phi _0), \dots , (t_p, \phi _p))$ satisfying that $t_0 < \dots < t_p$ and that the embeddings $\phi _i$ have disjoint cores; i.e. the sets $\phi _i(C)$ are disjoint.

(iii)

Topologise $K_p(W)$ using the $C^\infty$-topology on the space of embeddings, and let $K_p^\delta (W)$ be the same set considered as a discrete topological space.

(iv)

The assignments $[p] \mapsto K_p(W)$ and $[p] \mapsto K_p^\delta (W)$ define semisimplicial spaces, where the face map $d_i$ forgets $(t_i,\phi _i)$.

(v)

Let $K^\delta (W)$ be the simplicial complex with vertices $K^\delta _0(W)$, and where the (unordered) set $\{(t_0,\phi _0), \dots , (t_p,\phi _p)\}$ is a $p$-simplex if, when written with $t_0 < \dots < t_p$, it satisfies $((t_0, \phi _0), \dots , (t_p, \phi _p)) \in K_p^\delta (W)$.

We shall often denote a vertex $(t,\phi )$ simply by $\phi$, since $t$ is determined by $\phi$. Since a $p$-simplex of $K_\bullet ^\delta (W)$ is determined by its (unordered) set of vertices, there is a natural homeomorphism $|K_\bullet ^\delta (W)| \cong |K^\delta (W)|$.

We wish to associate to each simply connected $2n$-manifold a quadratic module with form parameter $(1, \{0\})$ if $n$ is even and $(-1, 2\mathbb{Z})$ if $n$ is odd. Essentially, we take the group of immersed framed $n$-spheres in $W$, with pairing given by the intersection form, and quadratic form given by counting self-intersections. We shall often have to work with framings, and for a $d$-manifold $M$ we let $\mathrm{Fr}(M)$ denote the frame bundle of $M$, i.e. the (total space of the) principal $GL_d(\mathbb{R})$-bundle associated to the tangent bundle $TM$.

In the following definition we shall use the standard framing $\xi _{S^n \times D^n}$ of $S^n \times D^n$, induced by the embedding

$$\begin{equation} \begin{aligned} S^n \times D^n &\longrightarrow \mathbb{R}^{n+1} \times \mathbb{R}^{n-1} = \mathbb{R}^{2n},\\ (x; y_1, y_2, \ldots , y_n) &\longmapsto (2e^{-\frac{y_1}{2}} x ; y_2, y_3, \ldots , y_n). \end{aligned} {\tag{5.1}} \end{equation}$$

This standard framing at $(0,0,\ldots ,0,-1;0) \in S^n \times D^n$ gives a point $b_{S^n \times D^n} \in \mathrm{Fr}(S^n \times D^n)$.

Definition 5.2.

Let $W$ be a compact manifold of dimension $2n$, equipped with a framed basepoint, i.e. a point $b_W \in \mathrm{Fr}(W)$, and an orientation compatible with $b_W$.

(i)

Let $I^\mathrm{fr}_n(W) = I^\mathrm{fr}_n(W,b_W)$ denote the set of regular homotopy classes of immersions $i: S^n \times D^n \looparrowright W$ equipped with a path in $\mathrm{Fr}(W)$ from $Di(b_{S^n \times D^n})$ to $b_W$. Smale–Hirsch immersion theory Hir59, Section 5 identifies this set with the homotopy group $\pi _n(\mathrm{Fr}(W), b_W)$ of the frame bundle of $W$. The (abelian) group structure this induces on $I^\mathrm{fr}_n(W)$ corresponds to a connect-sum operation.

(ii)

Let $\lambda : I^\mathrm{fr}_n(W) \otimes I^\mathrm{fr}_n(W) \to H_n(W;\mathbb{Z}) \otimes H_n(W;\mathbb{Z}) \to \mathbb{Z}$ be the map which applies the homological intersection pairing to the cores of a pair of immersed framed spheres.

(iii)

Let$$\mu : I^\mathrm{fr}_n(W) \longrightarrow \begin{cases} \mathbb{Z}& \text{$n$ even}\\ \mathbb{Z}/2 & \text{$n$ odd} \end{cases}$$

be the function which counts (signed, if $n$ is even) self-intersections of the core of a framed immersion (once it is perturbed to be self-transverse).

Lemma 5.3.

For any smooth simply connected $2n$-manifold $W$ equipped with a framed basepoint $b_W$ the data $\mathsf{I}^\mathrm{fr}_n(W) = (I^\mathrm{fr}_n(W), \lambda , \mu )$ is a quadratic module with form parameter $(1, \{0\})$ if $n$ is even and $(-1, 2\mathbb{Z})$ if $n$ is odd.

Proof.

Let $E \to W$ be the $n$th Stiefel bundle associated to the tangent bundle of $W$; there is a map $\phi : \mathrm{Fr}(W) \to E$ given by sending a frame to its first $n$ vectors, and this determines a basepoint $\phi (b_W) \in E$. Choosing a framing $\xi$ of $T_{(0,0,\ldots , 0,-1)}S^n$, Wall Wal70, Section 5 uses Smale–Hirsch immersion theory to identify the homotopy group $\pi _n(E, \phi (b_W))$ with the set $I_n(W)$ of regular homotopy classes of immersions $i: S^n \looparrowright W$ equipped with a path in $E$ from $Di(\xi )$ to $\phi (b_W)$.

The homomorphism $\pi _n(\phi ) : \pi _n(\mathrm{Fr}(W), b_W) \to \pi _n(E, \phi (b_W))$ is thus identified with the map $u: I^\mathrm{fr}_n(W) \to I_n(W)$ which restricts an immersion of $S^n \times D^n$ to $S^n \times \{0\}$. (In Wall’s identification $I_n(W) \cong \pi _n(E)$ the element $0 \in \pi _n(E)$ corresponds to an embedding $S^n \hookrightarrow \mathbb{R}^{2n} \subset W$ with trivial normal bundle. Our choice 5.1 is compatible with this.)

Our functions $\lambda$ and $\mu$ factor through $u$, and Wall shows Wal70, Theorem 5.2 that $(I_n(W), \lambda , \mu )$ is a quadratic module with the appropriate form parameter.

■

We remark that the bilinear form $\lambda$ can in general be quite far from non-degenerate. Let us also remark that the basepoint $b_W \in \mathrm{Fr}(W)$ and the paths from $Di(b_{S^n \times D^n})$ are used only for defining the addition on the abelian group $I^\mathrm{fr}_n(W)$, neither $\lambda$ nor $\mu$ depends on these data.

For the manifold $H= ([-1,0] \times D^{2n-1}) \cup _{\{-1\} \times D^{2n-1}}W_{1,1}$ we choose the framed basepoint $b_H$ given by the Euclidean framing of $T_{(0,0)} ([-1,0] \times D^{2n-1})$, i.e. the framing induced by the inclusion $[-1,0] \times D^{2n-1} \subset \mathbb{R}^{2n}$. We define canonical elements $e,f \in I^\mathrm{fr}_n(H)$ in the following way. There are embeddings

$$\begin{equation} \begin{aligned} \bar{e} : S^n \times D^n &\longrightarrow W_{1,1} \subset S^n \times S^n \subset \mathbb{R}^{n+1} \times \mathbb{R}^{n+1},\\ (x,y) &\longmapsto \left(x; \tfrac{y}{2}, - \sqrt {1- \vert \tfrac{y}{2}\vert ^2}\right), \end{aligned} {\tag{5.2}} \end{equation}$$

and

$$\begin{equation} \begin{aligned} \bar{f} : S^n \times D^n &\longrightarrow W_{1,1} \subset S^n \times S^n \subset \mathbb{R}^{n+1} \times \mathbb{R}^{n+1},\\ (x,y) &\longmapsto \left(-\tfrac{y}{2}, - \sqrt {1- \vert \tfrac{y}{2}\vert ^2}; x\right), \end{aligned} {\tag{5.3}} \end{equation}$$

which are orientation preserving. These may be considered as embeddings into $H$.

The embedding $\bar{e}$, together with a choice of path in $\mathrm{Fr}(H)$ from $D\bar{e}(b_{S^n \times D^n})$ to $b_H$, defines an element $e \in I^\mathrm{fr}_n(H)$, and $\bar{f}$ similarly defines an element $f \in I^\mathrm{fr}_n(H)$. These elements satisfy

$$\begin{equation*} \lambda (e,e) = \lambda (f,f) = 0, \quad \quad \lambda (e,f)=1, \quad \quad \mu (e)=\mu (f)=0, \end{equation*}$$

and so determine a morphism of quadratic modules $\mathsf{H}\to \mathsf{I}^\mathrm{fr}_n(H)$. (This morphism depends on the choice of paths in $\mathrm{Fr}(H)$ made above and is used to define the map 5.4 below. Once that map has been used to prove that its domain is highly connected, this choice plays no further role. The ambiguity in this choice arises only at the end of the proof of Lemma 5.5.)

Remark 5.4.

For use in Section 7, let us point out that there exists a framing of $H$ which is homotopic to the standard framing on the images of $\bar{e}$ and $\bar{f}$, and extends the Euclidean framing on $\{0\} \times D^{2n-1} \subset [-1,0] \times D^{2n-1} \subset H$. To see this, note that the standard framings on $\bar{e}(S^n \times D^n)$ and $\bar{f}(S^n \times D^n)$ are homotopic when restricted to the (contractible) intersection of these subsets, as both embeddings are orientation preserving. This allows us to construct the framing on $W_{1,1} \subset H$, and because $[-1,0] \times D^{2n-1}$ is glued to $W_{1,1}$ along an orientation preserving map, this framing may be extended to one of $H$ agreeing with the Euclidean framing on $\{0\} \times D^{2n-1}$.

If $(W_0, b_{W_0})$ and $(W, b_W)$ are manifolds with framed basepoints, a morphism $(e_0, \sigma _0) : (W_0, b_{W_0}) \to (W, b_W)$ consists of a (codimension 0) embedding $e_0 : W_0 \hookrightarrow W$ and a path $\sigma _0 : De_0(b_{W_0}) \rightsquigarrow b_W \in \mathrm{Fr}(W)$. Such a morphism induces a homomorphism of quadratic modules $(e_0, \sigma _0)_* : \mathsf{I}^\mathrm{fr}_n(W_0) \to \mathsf{I}^\mathrm{fr}_n(W)$, since $\lambda$ and $\mu$ are computed by counting intersections which may be done in either manifold. Furthermore, if $(e_1, \sigma _1) : (W_1, b_{W_1}) \to (W, b_W)$ is another morphism such that $e_1$ is disjoint from $e_0$ (up to isotopy), then $(e_0, \sigma _0)_*$ and $(e_1, \sigma _1)_*$ have orthogonal images in $\mathsf{I}^\mathrm{fr}_n(W)$.

For a manifold $W$ with distinguished chart $c : (-1,0] \times \mathbb{R}^{2n-1} \hookrightarrow W$ we choose $b_W$ to be induced by $Dc$ and the Euclidean framing of $T_{(0,0)}((-1,0] \times \mathbb{R}^{2n-1})$. Then an embedding $\phi : H\hookrightarrow W$ representing a vertex of $K^\delta (W)$ has a canonical homotopy class of path $\sigma _\phi$ from $\phi \circ b_H$ to $b_W$ (as the manifolds $\phi ([-1,0] \times D^{2n-1})$ and $c((-1,0] \times \mathbb{R}^{2n-1})$ are both contractible and framed). Thus $\phi$ gives a hyperbolic submodule

$$\begin{equation*} \mathsf{H}\longrightarrow \mathsf{I}^\mathrm{fr}_n(H) \overset{(\phi , \sigma _\phi )_*}{\longrightarrow }\mathsf{I}^\mathrm{fr}_n(W), \end{equation*}$$

and disjoint embeddings give orthogonal hyperbolic submodules, which defines a map of simplicial complexes

$$\begin{equation} K^\delta (W) \longrightarrow K^a(\mathsf{I}^\mathrm{fr}_n(W)). {\tag{5.4}} \end{equation}$$

We will use this map to compute the connectivity of $\vert K_\bullet ^\delta (W)\vert = |K^\delta (W)|$. The proof of the following lemma, and its generalisation in Section 7 to the presence of tangential structures, makes essential use of the Whitney trick.

Lemma 5.5.

If $(W, b_W)$ as in Lemma 5.3 has dimension $2n \geq 6$ and is simply connected, then the space $|K^\delta _\bullet (W)|$ is $\lfloor \tfrac{\overline{g}(W)-4}{2} \rfloor$-connected.

Proof.

For brevity we shall just write $K^\delta \to K^a$ for the map 5.4 and write $\overline{g} = \overline{g}(\mathsf{I}^\mathrm{fr}_n(W))$. We have $\overline{g}(W) \leq \overline{g}$, so it suffices to show that $|K^\delta _\bullet (W)|$ is $\lfloor \tfrac{\overline{g}-4}{2} \rfloor$-connected.

Let $k \leq (\overline{g}-4)/{2}$, and consider a map $h: \partial I^{k+1} \to |K^\delta |$, which we may assume is simplicial with respect to some PL triangulation $\partial I^{k+1} \approx |L|$. By Theorem 3.2, the composition $\partial I^{k+1} \to |K^\delta | \to |K^a|$ is nullhomotopic and so extends to a map $\overline{h}: I^{k+1} \to |K^a|$. By Theorem 3.2, we also have $\mathrm{lCM}(K^a) \geq \lfloor \tfrac{\overline{g}-1}{2}\rfloor \geq k+1$, so by Theorem 2.4 we may find a triangulation $I^{k+1} \approx |K|$ extending $L$ such that the star of each vertex $v \in K \setminus L$ intersects $L$ in a single simplex, and we change $\overline{h}$ by a homotopy relative to $\partial I^{k+1}$ so that $\overline{h}(\mathrm{Lk}(v)) \subset \mathrm{Lk}(\overline{h}(v))$ for each vertex $v \in K \setminus L$. We will prove that $\overline{h}: I^{k+1} \to |K^a|$ lifts to a nullhomotopy $F : I^{k+1} \to |K^\delta |$ of $h$.

Choose an enumeration of the vertices in $K$ as $v_1, \dots , v_N$ such that the vertices in $L$ come before the vertices in $K \setminus L$. For each $v_i \in L$ the vertex $h(v_i) = F(v_i) \in K^\delta$ is given by an embedding $j_i: H\to W$. For $v_i \in K \setminus L$ we shall inductively pick lifts of $\overline{h}(v_i) \in K^a$ to a vertex $F(v_i) \in K^\delta$ given by an embedding $j_i: H\to W$ satisfying

(i)

if $v_i,v_k$ are adjacent vertices in $K$ with $k < i$, then $j_i(C) \cap j_k(C) = \emptyset$,

(ii)

for $v_i \in K \setminus L$ the core $j_i(C)$ is in general position with $j_k(C)$ for $k < i$.

Suppose $\overline{h}(v_1), \dots , \overline{h}(v_{i-1})$ have been lifted to $j_1, \dots , j_{i-1}$ satisfying (i) and (ii). Then $v_i \in K\setminus L$ gives a morphism of quadratic modules $\overline{h}(v_i) = \phi : \mathsf{H}\to \mathsf{I}^\mathrm{fr}_n(W)$ which we wish to lift to an embedding $j_i$ satisfying the two properties. The element $\phi (e)$ is represented by an immersion $x : S^n \times D^n \looparrowright W$, which has $\mu (x)=0$, along with a path in $\mathrm{Fr}(W)$ from $Dx(b_{S^n \times D^n})$ to $b_W$. As $W$ is simply connected and of dimension at least 6 we may use the Whitney trick as in Wal70, Theorem 5.2 to replace $x$ by an embedding $j(e) : S^n \times D^n \hookrightarrow W$. Similarly, $\phi (f)$ can be represented by an embedding $j(f) : S^n \times D^n \hookrightarrow W$, along with another path in $\mathrm{Fr}(W)$.

As $\lambda (\phi (e), \phi (f))=1$, these two embeddings have algebraic intersection number 1. We may again use the Whitney trick to isotope the embeddings $j(e)$ and $j(f)$ so that their cores $S^n \times \{0\}$ intersect transversely in precisely one point, and so we obtain an embedding of the plumbing of $S^n \times D^n$ and $D^n \times S^n$, which is diffeomorphic to $W_{1,1} \subset H$. We then use the framed path from $Dx(b_{S^n \times D^n})$ to $b_W$ to extend to the remaining $[-1,0]\times D^{2n-1} \subset H$, giving an embedding $j : H\hookrightarrow W$. Setting $j_i = j$ may not satisfy (i) and (ii), but after a small perturbation it will satisfy (ii).

It remains to explain how to achieve that $j_1, \dots , j_i$ satisfy (i). If $v_k \in \mathrm{Lk}(v_i)$ is an already lifted vertex (i.e. $k < i$), we must ensure that $j_i(C) \cap j_k(C) = \emptyset$. Since they are in general position and have algebraic intersection numbers zero (as $\overline{h}$ is simplexwise injective so $\{v_i,v_k\}$ maps to a 1-simplex in $K^a$), we may use the Whitney trick to replace $j_i$ by an embedding satisfying $j_i(C) \cap j_k(C) = \emptyset$. The necessary Whitney discs may be chosen disjoint from $j_m(C)$, for all $m < i$ such that $m \neq k$ and either $v_m \in \mathrm{Lk}(v_i)$ or $v_m \in K \setminus L$, again since all such $v_m$ are in general position with each other. Then the Whitney trick will not create new intersections, and after finitely many such Whitney tricks we will have $j_i(C) \cap j_k(C) = \emptyset$ whenever $v_k \in \mathrm{Lk}(v_i)$ and $k < i$, ensuring that the lifts $j_1, \dots , j_i$ satisfy (i) and (ii).

In finitely many steps, we arrive at a lift of $\overline{h}: I^{n+1} \to K^a$ to a nullhomotopy of $h: \partial I^{n+1} \to K^\delta$, as desired. (Strictly speaking, we may not have lifted the chosen nullhomotopy $\overline{h}: I^{k+1} \to |K^a|$. The data of an element in $I^\mathrm{fr}_n(W)$ includes a path in $\mathrm{Fr}(W)$. If $W$ is spin, there will be two choices of such paths, related by a “spin flip”, and we have only lifted the $\phi (e)$ and $\phi (f)$ up to spin flip. Thus instead of lifting $\overline{h}: I^{k+1} \to |K^a|$ we may have lifted another nullhomotopy, related to $\overline{h}$ by spin flips on some vertices.)

■

Finally, we compare $|K_\bullet ^\delta (W)|$ and $|K_\bullet (W)|$. The bisemisimplicial space in Definition 5.7 below will be used to leverage the known connectivity of $|K_\bullet ^\delta (W)|$ to prove the following theorem, which is the main result of this section.

Theorem 5.6.

If $W$ is a compact simply connected manifold of dimension $2n \geq 6$ equipped with a framed basepoint, then the space $|K_\bullet (W)|$ is $\lfloor \tfrac{\overline{g}(W)-4}{2}\rfloor$-connected.

The semisimplicial space $K_\bullet (W)$ has an analogue in the case $2n=2$, which has been considered by Hatcher–Vogtmann HV15 and shown to be highly connected.

Definition 5.7.

With $W$ and $c$ as in Definition 5.1, let $D_{p,q} = K_{p+q+1}(W)$, topologised as a subspace of $K_p(W) \times K^\delta _q(W)$. This is a bisemisimplicial space, equipped with augmentations

$$\begin{align*} D_{p,q} &\overset{\varepsilon }{\longrightarrow }K_p(W),\\ D_{p,q} &\overset{\delta }{\longrightarrow }K_q^\delta (W). \end{align*}$$

Lemma 5.8.

Let $\iota : K_\bullet ^\delta (W) \to K_\bullet (W)$ denote the identity map. Then

$$\begin{equation*} |\iota | \circ |\delta | \simeq |\varepsilon |: |D_{\bullet , \bullet }| \longrightarrow |K_\bullet (W)|. \end{equation*}$$

Proof.

For each $p$ and $q$ there is a homotopy

$$\begin{align*} [0,1] \times \Delta ^p \times \Delta ^q \times D_{p,q} \longrightarrow \Delta ^{p+q+1} \times K_{p+q+1}(W),\\ (r,s,t,x,y)) \longmapsto ((rs,(1-r)t),(x,\iota y)), \end{align*}$$

where we write $(x,y) \in D_{p,q} \subset K_p(W)\times K_q^\delta (W)$, $(x,\iota y) \in K_{p+q+1}(W) \subset K_p(W) \times K_q(W)$, and $(rs,(1-r)t) = (rs_0,\dots , rs_p,(1-r)t_0, \dots , (1-r)t_q) \in \Delta ^{p+q+1}$. These homotopies glue to a homotopy $[0,1] \times |D_{\bullet ,\bullet }| \to |K_\bullet (W)|$ which starts at $|\iota | \circ |\delta |$ and ends at $|\varepsilon |$.

■

Proof of Theorem 5.6.

Let us write $\overline{g} = \overline{g}(W) \leq \overline{g}(\mathsf{I}^\mathrm{fr}_n(W))$. We will apply Corollary 2.9 with $Z = K_p(W)$, $Y_\bullet = K_\bullet ^\delta (W)$, and $X_\bullet = D_{p, \bullet }$. For $z = ((t_0,\phi _0), \dots , (t_p,\phi _p)) \in K_p(W)$, we shall write $W_z \subset W$ for the complement of the $\phi _i(C)$. The realisation of the semisimplicial subset $X_\bullet (z) \subset Y_\bullet = K_\bullet ^\delta (W)$ is homeomorphic to the full subcomplex $F(z) \subset K^\delta (W)$ on those $(t, \phi )$ such that $\phi (C) \subset W_z$ and $t > t_p$. The map of simplicial complexes 5.4 restricts to a map

$$\begin{equation*} F(z) \longrightarrow K^a(\mathsf{I}^\mathrm{fr}_n(W_z)). \end{equation*}$$

We have $\mathsf{I}^\mathrm{fr}_n(W_z) \oplus \mathsf{H}^{p+1} \cong \mathsf{I}^\mathrm{fr}_n(W)$, so $\overline{g}(\mathsf{I}^\mathrm{fr}_n(W_z)) = \overline{g}(\mathsf{I}^\mathrm{fr}_n(W))-p-1$, and hence by Theorem 3.2 the target is $\lfloor \tfrac{\overline{g}-4-p-1}{2}\rfloor$-connected. The argument of Lemma 5.5 shows that $F(z)$ is also $\lfloor \tfrac{\overline{g}-4-p-1}{2}\rfloor$-connected.

By Proposition 2.8, the map $\vert \varepsilon _p\vert : \vert D_{p, \bullet }\vert \to K_p(W)$ is a Serre microfibration, and we have just shown that it has $\lfloor \tfrac{\overline{g}-p-5}{2}\rfloor$-connected fibres, so by Proposition 2.6 it is $\lfloor \tfrac{\overline{g}-p-3}{2}\rfloor$-connected. Since $\lfloor \tfrac{\overline{g}-p-3}{2}\rfloor \geq \lfloor \tfrac{\overline{g}-3}{2}\rfloor - p$, we deduce by Proposition 2.7 that the map $|D_{\bullet ,\bullet }| \to |K_\bullet (W)|$ is $\lfloor \tfrac{\overline{g} -3}{2} \rfloor$-connected. But up to homotopy it factors through the $\lfloor \tfrac{\overline{g}-4}{2}\rfloor$-connected space $|K_\bullet ^\delta (W)|$, and therefore $|K_\bullet (W)|$ is $\lfloor \tfrac{\overline{g}-4}{2}\rfloor$-connected, too.

■

Finally, we define the semisimplicial space which we will use in the following section.

Definition 5.9.

Let $\overline{K}_\bullet (W) \subset K_\bullet (W)$ denote the sub-semisimplicial space with $p$-simplices those tuples of embeddings which are disjoint. (Recall that in $K_\bullet (W)$ we only ask for the embeddings to have disjoint cores.)

Corollary 5.10.

The space $|\overline{K}_\bullet (W)|$ is $\lfloor \frac{\overline{g}(W)-4}{2}\rfloor$-connected.

Proof.

We may choose an isotopy of embeddings $\rho _t : H\to H$, defined for $t \in [0,\infty )$, which starts at the identity, eventually has image inside any given neighbourhood of $C$, and which for each $t$ is the identity on some neighbourhood of $C$. Precomposing with the isotopy $\rho _t$, any tuple of embeddings with disjoint cores eventually become disjoint. It follows that the inclusion is a levelwise weak homotopy equivalence.

■

6. Resolutions of moduli spaces

In the Introduction we defined $\mathscr{M}(X)$ as the classifying space $B\mathrm{Diff}_\partial (X)$ of the group of diffeomorphisms of $X$ fixing its boundary. In this section we will describe a specific point-set model for this classifying space, together with a simplicial resolution. We then use the spectral sequence arising from this resolution to prove Theorem 1.2.

Definition 6.1.

For a $2n$-manifold $X$ with boundary $P$ and collar $c : (-\infty ,0] \times P \hookrightarrow X$, and an $\varepsilon >0$, let $\mathrm{Emb}_\varepsilon (X, (-\infty ,0] \times \mathbb{R}^\infty )$ denote the space, in the $C^\infty$-topology, of those embeddings $e$ such that $e \circ c(t, x) = (t, x)$ as long as $t \in (-\varepsilon ,0]$, and let

$$\begin{equation} \mathcal{E}(X)= \operatorname *{colim}_{\varepsilon \to 0} \mathrm{Emb}_\varepsilon (X, (-\infty ,0] \times \mathbb{R}^\infty ). {\tag{6.1}} \end{equation}$$

The space $\mathcal{E}(X)$ has a (free) action of $\mathrm{Diff}_\partial (X)$ by precomposition, and we write

$$\begin{equation*} \mathcal{M}(X) = \mathcal{E}(X)/\mathrm{Diff}_\partial (X). \end{equation*}$$

Two elements of $\mathcal{E}(X)$ are in the same orbit if and only if they have the same image so as a set, $\mathcal{M}(X)$ is the set of submanifolds $M \subset (-\infty ,0] \times \mathbb{R}^\infty$ such that

(i)

$M \cap (\{0\} \times \mathbb{R}^\infty ) = \{0\} \times P$ and $M$ contains $(-\varepsilon ,0] \times P$ for some $\varepsilon > 0$,

(ii)

the boundary of $M$ is precisely $\{0\} \times P$,

(iii)

$M$ is diffeomorphic to $X$ relative to $P$.

(The underlying set of $\mathcal{M}(X)$ depends on the specified identification $\partial X \cong P \subset \mathbb{R}^\infty$.)

By BF81 the quotient map $\mathcal{E}(X) \to \mathcal{E}(X)/\mathrm{Diff}_\partial (X)$ has slices and hence is a principal $\mathrm{Diff}_\partial (X)$-bundle. Since