Contractible polyhedra in products of trees and absolute retracts in products of dendrites

We show that a compact n-polyhedron PL embeds in a product of n trees if and only if it collapses onto an (n-1)-polyhedron. If the n-polyhedron is contractible and n\ne 3 (or n=3 and the Andrews-Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding. In contrast, there exists a 2-dimensional compact absolute retract X such that X\times I^k does not embed in any product of 2+k dendrites for each k.


Introduction
All spaces shall be assumed to be metrizable. By a compactum we mean a compact metrizable space. A finite-dimensional compactum is an ANR if and only if it is locally contractible; and an absolute retract (AR) if and only if it is a contractible ANR (see [5]). A one-dimensional compact AR is called a dendrite, and a one-dimensional compact ANR is called a local dendrite. An arbitrary connected one-dimensional compactum is sometimes called a curve. Theorem 1.1 (Nagata-Bowers [39], [7]; see also [46], [47], [2]). Every n-dimensional compactum X embeds in D n × I, where D is a certain dendrite.
It is well-known that every dendrite embeds in the 2-cube I 2 ; thus Theorem 1.1 may be viewed as an improvement of the classical Menger-Nöbeling-Pontriagin theorem that every n-dimensional compactum embeds in the (2n + 1)-cube I 2n+1 .
Theorem 1.1 is trivial in the case where X is a polyhedron: Theorem 1.2. Every compact n-polyhedron embeds in a product of n trees and I.
Proof. Given a triangulation K of the given polyhedron X, let S i be the set of all vertices of the barycentric subdivision K ′ that are barycenters of i-simplices of K. The simplicial map K ′ → S 0 * · · · * S n is clearly an embedding. Hence X embeds in I * S, where S = S 1 * · · · * S n . Next, I * S = pt * CS is homeomorphic to pt * (CS ∪ S × I) = I × CS. Finally, the cone CS is homeomorphic to the product of n trees CS 1 × . . . × CS n .
The above argument yields an explicit embedding of every compact n-polyhedron in I 2n+1 , which we have not seen in the literature. This is strange, for a part of this construction is certainly well-known; it yields The first author is supported by Russian Foundation for Basic Research Grant No. 11-01-00822, Russian Academy of Sciences program "Mathematical methods of construction and analysis of models of complex systems", Russian Government project 11.G34.31.0053 and Federal Program "Scientific and scientificpedagogical staff of innovative Russia 2009-2013". Proposition 1.3 ([17], [24]). The cone over every compact n-polyhedron embeds in a product of n + 1 trees. Theorem 1.4 (Borsuk-Patkowska [6]). The n-sphere S n does not embed in any product of n dendrites, for each n ≥ 0.
Another proof of Theorem 1.4 is given by the easy part of our Theorem 1.9 below. Theorem 1.5 (Gillman-Matveev-Rolfsen [17]). Every compact connected PL n-manifold with nonempty boundary embeds in a product of n trees.
This was originally a consequence of Proposition 1.3 along with a "reconstruction theorem" announced in [17]. Another proof of Theorem 1.5 is given by our Theorem 1.9 below, albeit the trees that it produces need not be cones over finite sets.
Nagata's original motivation for considering embedding into products of 1-dimensional spaces related to dimension theory (see [39]). Borsuk's proof of the 2-dimensional case of Theorem 1.4 was a solution to Nagata's problem; on the other hand, the author learned from W. Kuperberg, a student of Borsuk who has generalized Theorem 1.4 [29], that Borsuk saw this result as a part of his work on the problem of uniqueness of decomposition of ANRs into products. Yet another motivation for embedding into products of trees was the Poincaré Conjecture (now also known as Perelman's Theorem): Theorem 1.6. (a) (Gillman [16]) If a compact acyclic 3-manifold embeds in the product of a tree and I 2 , then it is collapsible.
(b) (Zhongmou [54]) Every compact connected 3-manifold with nonempty boundary embeds in the product of two triods and I.
A discussion of further results in the theory of embeddings into products of dendrites (or curves) can be found in the recent paper [24], which itself is a significant addition to this theory (see also additional details in [25]). We should mention Theorem 1.7 (Koyama-Krasinkiewicz-Spież [24]). There exists a 2-polyhedron that collapses onto a product of two graphs but does not embed in any product of two graphs. Yet it embeds in a product of two curves.
The 2-polyhedron in question is Θ × Θ ∪ J=I×{0} I × I, where Θ is the suspension over the three-point set, and the arc J lies in the interior of a 2-cell of Θ × Θ apart from one endpoint, which lies in a "corner" of that 2-cell.

1.A. Embedding contractible polyhedra in products of trees
Theorem 1.8. Every collapsible compact n-polyhedron PL embeds in a product of n trees. Moreover, the product of trees collapses onto the image of the embedding.
The embeddability in the 2-dimensional case is due to Koyama, Krasinkiewicz and Spież [24]. The principal additional ingredient in our proof of the general case is the Fisk-Izmestiev-Witte lemma [15; Lemma 57], [22], [52] (see also [19;Lemma 3.1], [10]), which asserts that for every finite set C (the 'palette') of cardinality #C ≥ d + 1, every C-colored combinatorial (d − 1)-sphere is color-preserving isomorphic to the boundary of a C-colored combinatorial ball. (A simplicial complex is C-colored if its vertices are colored by the elements of C so that no edge connects two vertices of the same color.) In particular, this lemma implies that if a triangulation of S 2 admits a 4-coloring, then it extends to a triangulation of the 3-ball where the link of every interior edge is (combinatorially) an even-sided polygon. As observed by R. D. Edwards and others in 1970s, the converse to this also holds: every such triangulation of the 3-ball has a 4-colorable boundary (see references in [22]).
The 2-dimensional case of Theorem 1.8 involves only the trivial case d ≤ 1 of the Fisk-Izmestiev-Witte lemma. Corollary 1.9. Let P be a compact n-polyhedron. The following are equivalent: (i) P PL embeds in a product of n trees; (ii) P PL embeds in a product of an (n − 1)-polyhedron and a tree; (iii) P collapses onto an (n − 1)-polyhedron; (iv) P PL embeds in a collapsible compact n-polyhedron.
Here (iv)⇒(i) follows from Theorem 1.8, (i)⇒(ii) is obvious, (ii)⇒(iii) is easy (see below), and to see that (iii)⇒(iv) it suffices to note that if P collapses onto Q then the amalgamated union P ∪ Q CQ is collapsible, where CQ is the cone over Q.
Proof of (ii)⇒(iii). Suppose that P is embedded in R × T , where R is an (n − 1)polyhedron and T is a tree, and P does not collapse onto any (n − 1)-polyhedron. Let P 0 be a triangulation of P . Then P 0 collapses onto a (generally non-unique) simplicial complex Q 0 that does not collapse onto any proper subcomplex. Then Q 0 has no free faces, and it follows that Q := |Q 0 | does not collapse onto any proper subpolyhedron. By the hypothesis Q is of dimension precisely n. The projection f : Q ⊂ R × T → R can be triangulated by a simplicial map Q 1 → R 1 . Let p is a point in the interior U of a top-dimensional simplex of R 1 such that the corresponding fiber F := f −1 (p) is of dimension precisely 1. The projection F ⊂ R × T → T is an embedding, so F is a forest. Thus F collapses onto a finite set, but is not a finite set itself; so it must have a free vertex. On the other hand,  [24]). An acyclic compact 2-polyhedron P embeds in a product of two trees if and only if P is collapsible. Remark 1.11. Let P be a compact polyhedron with H 1 (P ) = 0. If P embeds in a product of n graphs then it embeds in a product of n trees, namely in the product of (appropriate compact subtrees of) the universal covers of the n graphs. Thus "trees" can be replaced with "graphs" in Corollary 1.10 in accordance with [24]. (In fact, it was shown in [24] that an acyclic non-collapsible compact 2-polyhedron does not embed in any product of two curves.) Corollary 1.12. Let P be an n-polyhedron. For n = 3, the following are equivalent: (i) some product of n trees collapses onto a PL copy of P ; (ii) P collapses onto a contractible (n − 1)-polyhedron; (iii) some collapsible compact n-polyhedron collapses onto a PL copy of P . For n = 3, the same holds with "contractible" replaced by "3-deformable to a point".
A polyhedron P is said to be n-deformable to a polyhedron Q if they are related by a sequence of collapses and expansions (i.e. the inverses of collapses) through polyhedra of dimensions ≤ n. The Andrews-Curtis Conjecture asserts that all contractible 2polyhedra 3-deform to a point (see [1], [33]). Among its motivations (cf. Curtis [13; §2]) we mention that it would imply 1 that every contractible 2-polyhedron PL embeds in I 4 .
Proof. (iii)⇒(i) follows from Theorem 1.8 and (i)⇒(ii) follows from Corollary 1.9. To prove (ii)⇒(iii), suppose that P collapses onto an (n − 1)-polyhedron Q, and either Q is contractible, or n = 3 and Q 3-deforms to a point. Then by a result of Kreher-Metzler and Wall, there exists an (n − 1)-polyhedron R such that R collapses onto a PL copy of Q and R×I is collapsible [28; Satz 1a, Satz 1] (see also [1;§XI.4] for an outline of Kreher and Metzler's proof in English). Let S be the amalgamated union P ∪ Q=Q×{0} R × I. Then S ց R × I ց pt and S ց P ∪ Q R ց P . Remark 1.13. For each n ≥ 3 it is easy to construct a non-collapsible n-polyhedron that collapses onto a contractible (n − 1)-polyhedron (e.g. I n ∨ cone(f ) will do, where f is any degree 0 PL surjection S n−2 → S n−2 ). A more interesting example is due to M. M. Cohen, who constructed for each n ≥ 4 a contractible (n − 1)-polyhedron Q such that Q × I is not collapsible [12]. Other constructions (with very different proofs) are now known: P × I k−2 is not collapsible if P is the suspension over a (k − 1)-dimensional spine of a non-simply-connected homology k-sphere [3], and P × I q is not collapsible if P is a certain "(3q + 6)-dimensional dunce hat" [8].
A free deformation retraction of a space X onto a subspace Y is a homotopy h t : X → X starting with h 0 = id, ending with a retraction h 1 of X onto Y , and such that h t h s = h max(s,t) for all s, t ∈ [0, 1]. A space is freely contractible if it freely deformation retracts onto a point. Collapsibility is known to be strictly stronger than topological collapsibility [3], [8] and consequently than free contractibility; however, in the case of 2-polyhedra the three notions are equivalent [21]. Remark 1.15. The proof of Theorem 1.8 involves a (non-straightforward) construction of a collapsible cubulation of the given collapsible polyhedron, which might be of interest in its own right. Another such construction (a more straightforward one) has been used to characterize collapsible polyhedra in the language of abstract convexity theory [48], and to establish the 'only if' part of Isbell's conjecture: a compact polyhedron is collapsible if and only if it is injectively metrizable [30], [49; Chapter VI]. (Isbell himself proved that the two conditions are equivalent for 2-polyhedra [21].)

1.B. Embedding absolute retracts in products of dendrites
for some (or equivalently, every) metric on X, it admits an ε-map into Y for each ε > 0. We refer to [42] for a definitive discussion of the (quite subtle) difference between embeddability and quasi-embeddability of compact polyhedra in I m .
Our paper was originally motivated by the following problem. This problem appears as Problem 1.4 in [25] with the following comments: "Our next problem is of great interest, we believe it has affirmative solution." In the present paper, we shall prove Theorem 1.17. There exists a 2-dimensional compact AR X such that X × I k quasiembeds in a product of 2 + k dendrites but does not embed in any product of 2 + k curves, for each k ≥ 0.
The proof of the higher-dimensional (i.e. k ≥ 1) case is similar to (and only three lines longer than) the proof of the two-dimensional case. Similar arguments show that the Cartesian power X k quasi-embeds in a product of 2k dendrites, but does not embed in any product of 2k curves. Remark 1.18. A few months after we shared our proof of the two-dimensional case of Theorem 1.17 with J. Krasinkiewicz and S. Spież, they found their own solution of Problem 1.16 [27]. Compared to ours, it is amazingly simple (modulo their previous work with A. Koyama) -at least when slightly modified as follows.
The dunce hat D [53] (also known as the Borsuk tube [4], [27]) is easily seen to be the quotient of a collapsible polyhedronD by its only free edge. Indeed, the link L of the 0-cell e 0 of D is homeomorphic to . Let π : L → I be the projection. The star S of e 0 in some triangulation of D is homeomorphic to the cone over L, which can be viewed as the mapping cylinder of the constant map L → pt; we defineD by replacing S with the mapping cylinder MC(π). The target space I of π is identified with a free edge J inD, and clearlyD is collapsible.
The quotient mapD →D/J = D, being cell-like, is an ε-homotopy equivalence for each ε > 0 by Chernavsky's lemma [23; Lemma 1]; in particular, for each ε > 0 there where h is a homeomorphism such that h −1 (L * pt) lies in the ε 2 -neighborhood of e 0 , and g combines the quotient map L × I → MC(π) with a null-homotopy L * pt → I of π.) SinceD is collapsible, it embeds in a product of two trees ( [24]; see Corollary 1.10 above), so D quasi-embeds there; on the other hand, D does not embed in any product of two curves since it is contractible but not collapsible ( [24]; see Remark 1.11 above).
As observed in [27], similar arguments show that the Cartesian power D k quasiembeds in a product of 2k trees, but does not embed in any product of 2k curves. (This uses the more general result of [24] that no polyhedron P with rk H 1 (P ) < n and H n (P, P \ {x}) = 0 for each x ∈ P embeds in a product of n curves.) Remark 1. 19. Zeeman showed that D × I is collapsible [53], where D is the dunce hat. Hence D × I embeds in a product of 3 trees by Theorem 1.8. So the absolute retract X in Theorem 1.17 cannot be replaced by D. Moreover, it cannot be replaced by any 2-polyhedron R, since R × I embeds in a product of 3 trees by Proposition 1.3. Conjecture 1.20. (a) If a compact n-polyhedron P quasi-embeds in a product of n dendrites, then P × I embeds in a product of n + 1 trees.
Theorem 1.17 should be compared with the following results.
and k > 0, then the following are equivalent: (i) X embeds in I m ; (ii) X × I k embeds in I m+k ; (iii) X × T k embeds in I m+2k , where T denotes the triod.
In conclusion we note that the proof of non-embeddability in Theorem 1.17 involves the same kinds of local geometry and local algebra as that in the following Theorem 1.22 ). For each n > 1 there exists a compact ndimensional ANR, quasi-embeddable but not embeddable in I 2n .

Collapsible polyhedra
We use the following combinatorial notation [35; Chapter 2]. Given a poset P and a σ ∈ P , the cone ⌈σ⌉ is the subposet of all τ ∈ P such that τ ≤ σ, and the dual cone ⌊σ⌋ is the subposet of all τ ∈ P such that τ ≥ σ. The link lk(σ, P ) is the subposet of all τ ∈ P such that τ > σ, and the star st(σ, P ) is the subposet of all ρ ∈ P such that ρ ≤ τ for some τ ∈ ⌊σ⌋. If K is a simplicial complex (viewed as a poset of nonempty faces ordered by inclusion), and σ ∈ K, then lk(σ, K) is a simplicial complex, and st(σ, K) is isomorphic to ⌈σ⌉ * lk(σ, K). 2 Here the join is defined as follows. The dual cone C * P of the poset P consists of P together with an additional element0 that is set to be less than every element of P . The coboundary ∂ * Q of a dual cone Q = C * P , is the original poset P . (Note the relation with coboundary of cochains.) The product P × Q of two posets consists of pairs (p, q), where p ∈ P and q ∈ Q, ordered by (p, q) The canonical subdivision P # is the poset of all order intervals of P , ordered by inclusion. If K is a simplicial complex, then (C * K) # is a cubical complex. Conversely, if Q is a cubical complex and q ∈ Q, then lk(q, Q) is a simplicial complex, and st(q, Q) is isomorphic to ⌈q⌉ × (C * lk(q, Q)) # . Moreover, lk((p, q), P × Q) is isomorphic to lk(p, P ) * lk(q, Q). The details can be found in [35; Chapter 2]. Theorem 1.8 now follows from Lemma 2.1. Let K ց L be a simplicial collapse of simplicial complexes and let T 1 , . . . , T n be trees, so that Proof. Arguing by induction, we may assume that K ց L is an elementary simplicial collapse. Let Q denote the subcomplex of T cubulating f (|L|), and let B be the subcomplex of Q cubulating the image of the topological frontier of |L| in |K|. We may now forget K, L and f , remembering only that |B| is a PL ball of some dimension k < n.
We thus want to construct treesT i ⊃ T i and a subcomplex β ofT 1 × . . . ×T n such that β ∩ Q = B and |β| is a PL (k + 1)-ball. The boundary of |B| is cubulated by a subcomplex ∂B of B. Given a face q = q 1 × . . . × q n of B \ ∂B, we have lk(q, T ) ≃ lk(q 1 , T 1 ) * · · · * lk(q n , T n ). Each q i is either a vertex or an edge, and then lk(q i , T i ) is either a finite set or the empty set, accordingly. Let C be set of those i for which q i is a vertex. Then the cube ⌈q⌉ is of dimension n−#C, and consequently the dimension d − 1 of lk(q, B) equals k − n + #C − 1 < #C − 1.
Every vertex v of lk(q, T ) lies in lk(q i , T i ) for some i ∈ C; in that case let us color v by the ith color. In particular, the subcomplexes Λ := lk(q, Q) and S := lk(q, B) of lk(q, T ) are C-colored. Since #C > d, by the Fisk-Izmestiev-Witte lemma, the Ccolored combinatorial (d − 1)-sphere S bounds (abstractly) a C-colored combinatorial ball D. Let Λ + be the amalgamated union Λ ∪ S D, that is, the pushout of the diagram Λ ⊃ S ⊂ D in the category of C-colored simplicial complexes and color-preserving simplicial maps.
If D \ S contains k i vertices of color i, we define a new tree T + i = T i ∪ (q i * [k i ]) by attaching k i new edges to T i at the vertex q i for each i ∈ C (note that [k i ] = ∅ and so notion of link in Combinatorial Topology of 1960s was something slightly different: being defined only when P is a simplicial complex, it is canonically isomorphic to our lk(σ, P ) but is not identical with it.
The C-coloring of the vertices of lk(q, T ) extends to the similarly defined C-coloring of the vertices of lk(q, T + ). Then any color-preserving identification of the vertices of D \ S with the vertices of lk(q, T + ) that are not in lk(q, T ) extends uniquely to a color-preserving simplicial map Λ + → lk(q, T + ) that extends the inclusion Λ ⊂ lk(q, T ). This simplicial map is injective on vertices, hence is an embedding. By construction, Λ + ∩ lk(q, T ) = Λ.
In order to fit the above process in an inductive argument, let us now write Q 0 , B 0 for the given Q, B. Assuming that Q i , B i have been constructed, along with some distinct q 1 , . . . , q i ∈ (B 0 \ ∂B 0 ) \ B i , we repeat the above process with Q = Q i and B = B i , with one modification: q is now not an arbitrary face of B i \ ∂B i , but one that is also a face of the original B 0 \ ∂B 0 . Since q is still required to be a face of B i , our hypothesis entails that q / ∈ {q 1 , . . . , q i }. We set Q i+1 = Q + , B i+1 = B + , and q i+1 = q. Then q 0 , . . . , q i+1 ∈ (B 0 \ ∂B 0 ) \ B i+1 , which completes the inductive step. Since B 0 \ ∂B 0 is finite, the number of steps is bounded. If the final step is rth, it is easy to see that B r ∩ B 0 = ∂B 0 = ∂B r , and B 0 ∪ B r bounds a cubical combinatorial (k + 1)-ball β (namely, β is the union of all the (k + 1)-balls of the form E) such that β ∩ Q 0 = B 0 and β ∪ Q 0 = Q r .
Remark 2.2. The combinatorial type of the ball β depends on the order in which q 1 , . . . , q r are picked out of B 0 \ ∂B 0 . For instance, suppose that n = 2, k = 1 and the arc B 0 consists of e edges (and hence e + 1 vertex). If e > 1, then we may take q 1 , . . . , q r to be all the non-boundary vertices, ordered consecutively, which will lead to the same β as in [24]. For instance if e = 2 (so r = 1) and T 1 = Q 0 = B 0 , T 2 = pt, thenT 1 = T 1 ,T 2 is a single edge, and Q r = B r =T 1 ×T 2 (which amounts to two squares). On the other hand, if we first pick out all the edges (in any order) and then the e − 1 non-boundary vertices (in any order), the result will be unique, but quite different from the above. For instance if e = 2 (so r = 3) and T 1 = Q 0 = B 0 , T 2 = pt, then at the final stepT 1 is a triod,T 2 contains two edges, and B r consists of four squares. Picking out only vertices but not consecutively may also lead to a β different from that in [24]. Remark 2.3. As discussed in the previous remark, the construction in the proof of Lemma 2.1 depends on the choices of the cubes q 1 , . . . , q r . Let us describe a canonical range of choices that all lead to the same embedding. Each tree T i is constructed in stages pt = T i0 ⊂ · · · ⊂ T is = T i . The vertices of T i are partially ordered by v < w if there exists a k < s such that v ∈ T ik and w / ∈ T ik , yet w and v belong to the same component of |T i \ ⌊T i,k−1 ⌋|. (In particular, incomparable vertices are non-adjacent in the tree.) This yields a partial order on the vertices of B \ ∂B ⊂ Q ⊂ T 1 × . . . × T n . Let q 1 , . . . , q r be the vertices of B \ ∂B arranged in some total order extending the constructed partial order. It is clear then that r is indeed the last stage of the construction, and that Q r does not depend on the choice of the total order.
An alternative proof of the implication (iii)⇒(i) in Theorem 1.9 is given by Lemma 2.1 along with the following lemma (take k = n − 1). The prejoin P + Q consists of the elements of P ∪ Q with the order defined as follows: p q iff either p, q ∈ P and p ≤ q in P ; or p, q ∈ Q and p ≤ q in Q; or p ∈ P and q ∈ Q. Note that C * P ≃ pt + P . It is easy to see that (P + Q) ♭ ≃ P ♭ + Q ♭ , where P ♭ denotes the barycentric subdivision (see details in [35]).

Local cohomology
By H * we denote the Alexander-Spanier cohomology [45], [31], or equivalently (see [44]) sheaf cohomology with constant coefficients [9]. If the coefficients are omitted, they are understood to be integer. The case of coefficients in a field is much easier (see [50]) but will not suffice for our purposes.
If (X, Y ) is a pair of compacta, H i (X, Y ) is isomorphic to the direct limit lim where · · · → (P 1 , Q 1 ) → (P 0 , Q 0 ) is any inverse sequence of pairs of compact polyhedra with inverse limit (X, Y ). In particular, every cohomology group H i (Y, X) is countable. More generally, when Y is closed in X (which we always assume to be metrizable), then H i (X, Y ) coincides (see [44]) with theČech cohomology of (X, Y ), which may be defined as the direct limit of the ith cohomology groups of the nerves of all open coverings of (X, Y ). In particular, if Y is closed in X and X is n-dimensional, then H i (X, Y ) = 0 for i > n (since covers with at most n-dimensional nerve form a cofinal subset in the directed set of all open covers of X).
If X is a compactum and x ∈ X, the local cohomology group H i (X, X \ {x}) is isomorphic to lim such that U k = {x} and each Int U k ⊃ Cl U k+1 . As observed in [43; §1], this follows from the exact sequences of the pairs (U k , U k \ {x}) and the fact that the direct limit functor preserves exactness of sequences. However, this isomorphism will not be used in the sequel.
Instead, we shall use the following more geometric description of the local cohomology groups (parallel to [38; proof of Lemma 1]).
Hence from the preceding discussion we obtain Proof of Proposition 3.1. We shall show that (X, X \ {x}) is "almost" homotopy equivalent to the mapping telescope of pairs (X, X \ U i ), meaning that there is a map of pairs in one direction, which admits a homotopy inverse separately on each entry of the pair; by the Five Lemma, this is just good enough as long as cohomology is concerned.
The ) . It is easy to see that g is homotopy inverse to the restriction h : X ×[0, ∞)\U [0,∞) → X \{x} of the projection X ×[0, ∞) → X; hence h is a homotopy equivalence. Using the isomorphisms induced by g and the homotopy equivalence X × [0, ∞) → X, the Five Lemma implies that f * is an isomorphism.
By well-known arguments (see [37; proof of Theorem 4] or [34; proof of equation ( * ) in §1.B or proof of Theorem 3.1(b)]), Proposition 3.1 gives rise to a Milnor-type natural short exact sequence (found explicitly in [18]): if X is an n-dimensional compactum. Proof. Let U k be open neighborhoods of x in X as in Proposition 3.1. The restriction map H n (X, But by naturality of the isomorphism ( * ), lim Z (this occurs in Remark 3.5 and is called "Jacob's ladder" in [20]). Example (i) cannot occur in ( * ) with n = 1, because there is "not enough room for twisting" in one-dimensional spaces, so we cannot expect to find even a single multiplication as in (i). On the other hand, if X is an LC n compactum, then we cannot find example (ii) in ( * ), because n-cohomology of compact subsets of X is "almost" finitely generated in the sense that for every two compact subsets K ⊂ X and L ⊂ Int K, the image of H n (K) → H n (L) is finitely generated [9; II.17.5 and V.12.8].
Consider a composition f : By the naturality of the universal coefficients formula (see [9; V.12.8, V.13.7] The image of f * is a subgroup of the free abelian group H 0 (L j ). So it is itself free abelian, in particular, projective as a Z-module. Hence f * is a split epimorphism onto its image. Then the inclusion of the image of f * into H 0 (L i ) is a split monomorphism. (Indeed, given abelian group homomorphisms f * : G → H, f * : Hom(H, Z) → Hom(G, Z) defined by f * (ψ) = ψf * , and s : im f * → G such that f * sf * = f * , define r : Hom(G, Z) → im f * by r(ϕ) = ϕsf * ; then rf * = f * , i.e. r(ψf * ) = ψf * for each ψ ∈ Hom(H, Z).) Thus f * is a homomorphism onto a direct summand of H 0 (L i ). The finitely generated group H 0 (L i ) contains no infinitely decreasing chain of direct summands; so the inverse sequence · · · → H 0 (L 1 ) → H 0 (L 0 ) satisfies the Mittag-Leffler condition. Hence so does · · · → H 0 (K 1 ) → H 0 (K 0 ).
On the other hand, consider a composition g : is a subgroup of the free abelian group H 1 (L i ). So it is itself free abelian, in particular, projective as a Z-module. Hence g * is a split epimorphism onto its image. Then the kernel of g * is a direct summand in H 1 (X). The finitely generated group H 1 (X) contains no infinitely decreasing chain of direct summands; hence the homomorphisms H 1 (X) → H 1 (L i ) have the same kernel for all sufficiently large i. Then so do the homomorphisms H 1 (X) → H 1 (K i ). Since X is 1-dimensional, the latter are surjective. Hence H 1 (K i+1 ) → H 1 (K i ) are isomorphisms for sufficiently large i. In particular, · · · → H 1 (K 1 ) → H 1 (K 0 ) satisfies the dual Mittag-Leffler condition.

Skliarienko's compactum
We note that if the compactum X is the limit of an inverse sequence of compacta X i , all of which embed in Y , then X quasi-embeds in Y (for it follows from the definition of the topology of the inverse limit that the maps X p ∞ i − − → X i ⊂ Y are ε i -maps with respect to any fixed metric on X, where ε i → 0 as i → ∞). The converse implication (which we shall not need here) holds when Y is a polyhedron (a simple proof should appear in a future version of [36]; see also [32; Theorem 1] but beware that their "ε-maps" are required to be surjective).

Skliarienko's compactum.
Given a direct sequence X 1 → X 2 → . . . , the mapping telescope Tel(X 1 → X 2 → . . . ) is the infinite union MC(X 1 → X 2 ) ∪ X 2 MC(X 2 → X 3 ) ∪ X 3 . . . of the mapping cylinders (the direct limit of the finite unions). Let X be the one-point compactification of the mapping telescope of the direct sequence of two-fold coverings. It is easy to see that X is a contractible and locally contractible 2-dimensional compactum, and so an AR. It was introduced by Je. G. Skliarienko [43;Example 4.6]. We shall call X the Skliarienko compactum.
Proof. Let us represent X as an inverse limit of polyhedra. To this end, consider the following mapping telescope of a direct sequence: j stands for a copy of S 1 . Note that X contains the cone D 2 = Tel(S 1 i → pt). Let f i : X i+1 → X i be the composition of the quotient map X i+1 → X i+1 /D 2 and a homeomorphism X i+1 /D 2 → X i which is the identity on Tel(S 1 . Then X is homeomorphic to the inverse limit of . . . Notice that each X i is a collapsible 2-polyhedron. Hence by a result of Koyama, Krasinkiewicz and Spież (see Corollary 1.10), X i embeds in a product of two trees T i and T ′ i . Let us consider the cluster T = lim where the basepoint of each T i is one of its endpoints. Let T ′ be the analogous cluster of the trees T ′ i . Then T and T ′ are dendrites, T contains a copy of each T i , and T ′ contains a copy of each T ′ i . Thus each X i embeds in T × T ′ . Therefore X quasi-embeds there.
Let X be the Skliarienko compactum and let ∞ ∈ X denote the remainder point of the one-point compactification. It is easy to see that H 3 (X, X \ {∞}) is non-zero [43]. More generally, let us compute H 3+k (X × I k , X × I k \ {(∞, 0)}), where I = [−1, 1]. Let F i be the union of the first i mapping cylinders in the mapping telescope: i , and these collapses identify up to homotopy the inclusions F i ⊂ F i+1 with the two-fold coverings S 1 . By the Künneth formula (see references in the proof of Lemma 3.3), H 2+k (X × I k , G i ) ≃ H 2 (X, F i ), and the inverse sequence · · · → H 2+k (X × I k , G 2 ) → H 2+k (X × I k , G 1 ) is again of the form . . .   Proof. It is well-known that locally contractible compacta have finitely generated cohomology groups (see [9;II.17.7], [34; 6.11]). If a locally connected n-dimensional compactum X with finitely generated H n (X) embeds in a product n curves, then the first several lines of the proof of Theorem 2.B.1 in [24] (which contain further references) produce an embedding of X in a product of n local dendrites.   Remark 4.6. If · · · → G 1 → G 0 is an inverse sequence of countable groups, let lim ← 1 f g G i be the direct limit lim → L α of the derived limits L α = lim ← 1 H αi over all inverse sequences · · · → H α1 → H α0 of finitely generated subgroups H α i ⊂ G i , where the bonding maps are the restrictions of those in · · · → G 1 → G 0 . Some results about lim ← 1 f g will appear in a future paper by the first author. By using the functor lim Remark 4.7. The same arguments (only using the general case of Theorem 1.8 rather than the easier 2-dimensional case) show that the n-dimensional Skliarienko compactum (similarly defined with S n−1 in place of S 1 ) quasi-embeds in a product of n dendrites, but does not embed in a product of n curves.

Co-local contractibility
Let us call a compactum X co-locally contractible at x ∈ X if every neighborhood U of x contains a neighborhood V of x such that the inclusion X \ {x} ⊂ X is homotopic to a map X \{x} → X \V ⊂ X by a homotopy keeping X \U fixed. (Equivalently, every neighborhood U of x contains a neighborhood V of x such that for every neighborhood W of x contained in V , the inclusion X \ W ⊂ X is homotopic to a map X \ W → X \ V by a homotopy keeping X \ U fixed.) We call X co-locally contractible if it is co-locally contractible at every point. (Compare Borsuk's idea of colocalization [5; §IX.16] and colocal connectedness of Krasinkiewicz and Minc [26].) Remark 5.1. A slightly stronger property than co-local contractibility, obtained by replacing the inclusion X \ {x} ⊂ X with the identity map of X \ {x}, is known as reverse (or backward) tameness of X \ {x} (see [41], [20]). Dually, X \ {x} is called forward tame if there exists a closed neighborhood U of x such that for every neighborhood V of x, the inclusion V \{x} ⊂ X \{x} is properly homotopic to a map V \{x} → U \{x} ⊂ X \{x} (see [41], [20]). It is not hard to see (even if appears surprising) that forward tameness of X \ {x} implies local contractibility of X at x. To see that the converse implication fails, let P be the suspension of a non-contractible acyclic polyhedron and let its basepoint b be one of the two suspension points; or alternatively let P be the dunce hat and b its unique 0-cell. Then the cluster C = lim ← (· · · → P ∨ P ∨ P → P ∨ P → P ) of copies of P is an AR, yet it follows from Dydak-Segal-Spież [14] that C \ {b} is not forward tame.
Proof. This is a straightforward diagram chasing. The hypothesis implies that, with x, U and V as above and for each i, the restriction map H i (X \ {x}) → H i (X \ V ) is a split injection on the image of H i (X). Hence the image of the forgetful map f : H i (X \ {x}, X \ V ) → H i (X \ {x}) lies in the image of H i (X). The latter equals the kernel of the coboundary map δ : H i (X \ x) → H i+1 (X, X \ {x}), hence δf = 0. Since this δf : H i (X \ {x}, X \ V ) → H i+1 (X, X \ {x}) is also the coboundary map, the restriction H i+1 (X, X \ {x}) → H i+1 (X, X \ V ) must be an injection. Finally, since X is n-dimensional and without loss of generality V is open, H n+1 (X, X \ V ) = 0. Thus H n+1 (X, X \ {x}) = 0.