1. The general notion of a discriminant is as follows. Consider
any function space ${\scr F}$, finitedimensional or not, and some
class of singularities $S$, which the functions from ${\scr F}$ can
take at the points of the issue manifold. The corresponding
discriminant variety $\sum(S)$ is the set of all functions in ${\scr
F}$ having such singularities. For example, let ${\scr F}$ be the
space of polynomials of the form
$$x^d+a_1x^{d-1}+\cdots+a_d,\tag1$$
and $S=\{a multiple root\}$. Then $\sum(S)$ is the zero set of the
usual discriminant polynomial on coefficients $a_i$; this is a
motivation for the above general definition. In general, for any
$k\geq 2$ we can consider the discriminant $\sum_k$ which consists of
all polynomials having roots of multiplicity at least $k$.
Many famous topological spaces can be described as the complements of
appropriate discriminants (or at least are homotopy equivalent to
them). For instance, so are: the classifying spaces of braid groups;
classical Lie groups; spaces of Morse and generalized Morse functions,
or, more generally, spaces of maps $M\to {\bf R}^n$ without
complicated singularities; iterated loop spaces $\Omega^mS^n$, $m\leq
n$, or, more generally, the spaces of continuous maps $X\to Y$ where
$X$ is an $m$-dimensional cell complex, and $Y$ is an $m$-connected
one; the spaces of knots and linkts in ${\bf R}^n$, $n\geq 3$, and
many others.
Now, I describe a general spectral sequence which calculates the
cohomologies of all these spaces, and, in many cases, also the stable
homotopy types of them.
2.The key notion in this construction is the Spanier-Whitehead duality. (Two topological spaces are S-W dual if they have homotopy types of complementary subsets in a sphere of appropriate dimension: the homology groups of S-W dual spaces are related by the Alexander duality.) This duality is an involution on the space of stable homotopy types: the s.h. type of a topological space is determined by that of any of its S-W dual space.
3. For the simplest example, consider the space $P_d\sbs \sum_k$
of real polynomials of the form (1) having no roots of multiplicity
$\geq k$. Its S-W dual space is the one-point compactification
$\overline\sum_k$ of the discriminant variety $\sum_k$, and we can
replace the study of all stable properties of our space by that of
this compactification. (This reduction, due to Arnold (168, see ref.
$[Ar_4]$ in [1]) is very useful, because the space of nondiscriminant
objects is open and has no evident geometrical structure, while the
discriminant is a naturally stratified set, and a lot of its topology
can be expressed through this stratification.)
Further, we construct a geometrical resolution of
$\tilde\sum_k$, i.e., a space with the same homotopy type but with a
more explicit homological structure. First, we take the
tautological normalization of $\sum_k$, i.e., the space of all pairs
$\{x\in {\bf R}^1, f\in P_d\}$ such that $f$ has a root of
multiplicity $\geq k$ at $x$. This space is diffeomorphic to ${\bf
R}^{d-1}$, and forgetting the first elements of the pairs defines a
proper map of it onto $\sum_k$. The points of the preimage of any
polynomial $f$ correspond to all $\geq k$-fold roots of $f$. Then, for
any such $f\in \sum_k$ we insert in appropriate way a simplex, whose
vertices correspond to these preimage points. The union of all
inserted simplices is the desired geometrical resolution $\sigma_{\bf k}$ of
$\sum_k$. There is a natural proper projection $\sigma_{\bf k}\to
\sum)_k$, whose extension to the map of one-point compactifications is
a homotopy equivalence, and it is sufficient to study only the
compactification $\overline\sigma_{\bf k}$ of $\sigma_{\bf k}$. This
space has a natural filtration: its term $F_p$ consists of all
inserted simplices of dimension $\leq p-1$. The term $E^1_{p,q}$ of
the corresponding homological spectral sequence equals $\overline
H_{p+q}(F_p-F_{p-1})$, where $\overline H_*$ denotes the Borel-Moore
homology. On the other hand, the space $F_p-F_{p-1}$ has a natural
structure of a fibre bundle, whose base is the space ${\bf R}^1(p)$ of
all subsets of cardinality $p$ in ${\bf R}^1$, and the fiber is the
direct product of an open $(p-1)$-dimensional simplex and an affine
space of dimension $d-p\cdot k$. Thus $E^1_{p,q}$ equals ${\bf Z}$ if
$q=d-p(k-1)-1$ and $p\leq[d/k]$, and is trivial for all other $p,q$.
Obviously our sequence degenerates at this term $E^1$ and gives
immediately the answer.
It is the time now to remember that we are calculating not the
homology of the discrimnant, but the (Alexander dual to it) cohomology
of its complement. Therefore it is natural to invert our spectral
sequence formally into a cohomological one by setting $E^{p,q}_r\equiv
E^r_{-p,d-q-1}$, so that the resulting sequence lies in the second
quadrant and converges exactly to the cohomology of $P_d\sbs \sum_k$.
Its term $E^{p,q}_1$ equals ${\bf Z}$ if $q=-p(k-1)$, $p\in[-d/k],0]$,
and is trivial for other $p,q$.
4. An important property of this sequence is its stabilization when $d$ increases: for any $d'>d$, the corresponding sequences, calculating the cohomologies of spaces $P_d\sbs \sum_k,P_{d'}\sbs\sum_k$, coincide in the "stable" domain $\{p,q\colon p\geq -[d/k]\}$. These spaces $P_d\sbs \sum_k$ can be considered as finite dimensional approximations of a space of smooth functions ${\bf R}^1\to{\bf R}^1$ with, some fixed behavior at infinity and without zeros of multiplicity $\geq k$; our spectral sequences for increasing $d$ actually converge to a stable spectral sequence calculating the cohomology of the later space. Moreover, these sequences prove that this space is homology equivalent to the loop space $\Omega({\bf R}^{k-1}\sbs 0)\sim\Omega S^{k-1}$ (and this equivalence is defined by the jet extensions). Indeed, this is a very special case of the following general situation.
5. Spaces of functions without complicated singularities. Let
$S$ be a class of singularities of maps of $m$-dimensional manifolds
to ${\bf R}^n$, i.e., a semialgebraic ${\rm Diff}({\bf
R}^m)$-invariant closed subset in the space of $k$-jets of maps ${\bf
R}^m\to {\bf R}^n$. Then for any $m$-manifold $M$ the corresponding
subset $S(M)$ in the jet space $J^k(M,{\bf R}^n)$ is well-defined.
Theorem. If $M$ is a closed manifold and the set of maps $M\to
{\bf R}^n$ having the singularities of the class $S$ has codimension
$r\geq 2$ in the space of all smooth maps, then the space of maps
without such singularities is homology equivalent to the space of
continuous sections of the jet bundle $J^k(M,{\bf R}^n)\to M$ which do
not intersect $S(M)$; this equivalence is induced by the $k$-jet
extensions of maps.
(For several other results of this kind, see ref. $[I_1]$ in [1].)
Indeed, the homologies of both spaces can be calculated by the
spectral sequences generalizing the above one and naturally isomorphic
starting with the term $E_1$. This term lies in the edge $\{p,q\colon
p<0,rp+q\geq 0\}$ and has an explicit description: say, for any
negative $p$ the column $E^{p,*}_1$ equals (up to a shift and
inversion of dimensions) the Borel-Moore homology group of a natural
fibre bundle, whose base is the space $M(-p)$ of subsets of
cardinality $-p$ in $M$, and the fiber is the $(-p)$-th Cartesian
product of the set $S$.
6. The previous theorem has also complex analogues: say, the May-Segal formula for the homology of the stable braid group, $H^*({\rm Br}(\infty))\cong H^*(\Omega^2S^3)$, is of the same nature.
7. The simplest version of this spectral sequence, defined by
singularities of 0-jets, calculates the cohomology of {\bf spaces of
continuous maps of arbitrary $m$-dimensional $CW$-complex $X$ into
$m$-connected complex $Y$}. (The simplest version of this simplest
version is the Adams spectral sequence for the loop spaces.) Indeed,
we may assume that $Y$ is finite dimensional. Imbed it in a sphere
$S^N$ ($N$ very large), and let $\Psi$ be a S-W dual to $Y$ subcomplex
in $S^N$. Let $C\Psi$ be the union of rays in ${\bf R}^{N+1}$ issuing
from the origin and penetrating the unit sphere in the points of
$\Psi$. Take for the function space ${\scr F}$ the space of continuous
maps $X\to {\bf R}^{N+1}$, and define the discriminant as the set of
maps whose images intersect $C\Psi$. Since $Y$ is $m$-connected,
$\Psi$ can be chosen to be of codimension $\geq m+2$ in $S^N$, and
hence the codimension $r$ of the disriminant in ${\scr F}$ is $\geq
2$. The space of maps $X\to Y$ is obviously homotopy equivalent to the
complement of this discriminant, and hence the cohomology of this space
can be calculated by a spectral sequence very similar to the previous
one: it lies in the second quadrant in the edge $\{p,q\colon
p<0,rp+q\geq 0\}$, and its term $E^{p,q}_1$ is as follows. Let
$\Pi(t)$ be the fibre bundle over the configuration space $X(t)$,
whose fiber over a point $\{z_1,\cdots, z_t\}$ is the Cartesian
product of $t$ examples of $C\Psi$ which are in a correspondence with
the points $z_i$ (and are permuted in the obvious way over the loops
in the base space $X(t)$ permuting these points). Then,
$E^{p,q}_1\cong \overline H_{-p(N+1)-q}(\Pi(-p),(\pm{\bf Z})^{\otimes
N})$, where $\pm{\bf Z}$ is the local group locally isomorphic to
${\bf Z}$ and changing the orientation over the loops defining odd
permutations of the points $z_i$. This spectral sequence has also a
"relative" variant (calculating the homology of maps $X\to Y$
coinciding with a fixed map on some subcomplex $K$ in $X$): in this
case $\Pi(-p)$ in the previous formula should be defined as a similar
fibre bundle over $(X\sbs K)(-p)$.
If both $X$ and $Y$ are the spheres, this spectral sequence
degenerates at the first term and gives the Snaith splitting formula
for the homology of iterated loop spaces of spheres.
Also in the "marginal" case, when $X$ is $m$-dimensional and $Y$ is
$(m-1)$-connected, this spectral sequence provides some homology
classes of $Y^X$, but generally not all of them.
Here are few more consequences of the same technics.
8. Complements of resultant and discriminant varieties}. Consider
the space $(CP_d)^k$ of all systems of $k$ complex polynomials of the
form (1) in ${\bf C}^1$.
Theorem. The space of nonresultant (i.e., having no common
roots) systems in $(CP_d)^k$ is stable homotopy equivalent to the
space of all complex polynomials (1) of degree $dk$ without roots of
multiplicity $\geq k$. The same is true also for real systems and
polynomials.
(In the case $k=2$, this fact was established previously by F. R.
Cohen, R. L. Cohen, B. M. Mann and R. J. Milgram, see reference [CCMM]
in [1].)
9. We can define a discriminant as the space of all maps $S^1\to {\bf R}^n$, $n\geq 3$, which are not the knots, i.e., have singularities of selfintersections. Then, a spectral sequence similar to the above one provides many (all if $n\geq 4$) cohomology classes of the spaces of knots and links, in particular (if $n=3$) the knot and link invariants. A majority of the standard notions of the corresponding theory appears naturally in this spectral sequence: say, the groups $E^{-i,i}_0$ (resp., $E^{-i,i+1}_0$) are generated by the "chord diagrams" (resp., "4-term relations" and trivial relations of "vanishing loops").
10. Spaces of curves without triple intersections, spaces of nonsingular complex surfaces. splittings for the homology of Lie groups, etc.
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