ON THE HOMOTOPY INVARIANCE OF
CONFIGURATION SPACES
MOKHTAR AOUINA AND JOHN R. KLEIN
Abstract. For a closed PL manifold M, we consider the configuration
space F (M, k) of ordered k-tuples of distinct points in M. We show that
a suitable iterated suspension of F (M, k) is a homotopy invariant of M.
The number of suspensions we require depends on three parameters: the
number of points k, the dimension of M and the connectivity of M. Our
proof uses a mixture of embedding theory and fiberwise algebraic topolog*
*y.
1. Introduction
For a closed PL manifold M and an integer k 2, we will consider the
configuration space
F (M, k) := {(x1, ..., xk)| xi2 M and xi6= xj fori 6= j} .
A fundamental unsolved problem about these spaces concerns their homotopy
invariance: when M and N are homotopy equivalent, is it true that F (M, k)
and F (N, k) are homotopy equivalent?
Here is some background. It is known that the based loop space F (M, k),
is a homotopy invariant (see Levitt [L ]). When M is smooth, the cohomol-
ogy of F (M, k) with field coefficients has been intensively studied (see e.g.,
Bödigheimer-Cohen-Taylor [B-C-T ]). When M is a smooth projective variety
over C, Kriz [Kr ] has shown that the rational homotopy type of F (M, k) de-
pends only on the rational cohomology ring of M. These results indicate that
if homotopy invariance fails, a counterexample will be difficult to come by.
When k = 2 we have F (M, 2) = M x M - is the deleted product. Even
in this instance, the homotopy invariance question is still not settled (althou*
*gh
partial results are known; see Levitt [L ]).
The purpose of this paper is to show that a suitable iterated suspension of
F (M, k) is a homotopy invariant. The bound on the number of suspensions we
need to take depends on three parameters: the number of points, the dimension
of M and the connectivity of M.
____________
Date: October 30, 2003.
The second author is partially supported by NSF Grant DMS-0201695.
2000 MSC. Primary 55R80; Secondary 57Q35, 55R70.
1
2 MOKHTAR AOUINA AND JOHN R. KLEIN
For an unbased space Y , we define its j-fold suspension
jY := (* x Sj- 1) [ (Y x Dj) ,
where the union is amalgamated along Y x Sj- 1(up to homotopy, jY is the
join of Y and Sj- 1).
For integers d, k 3 and r 0, define
ff(k, d, r) := (k - 2)d - r + 2 .
When k = 2 and r 3, or when d 2, we set ff(k, d, r) := 0.
Our main result is
Theorem A. Let M and N be homotopy equivalent closed PL manifolds of
dimension d. Assume M is r-connected for some r 0. Then there is a
homotopy equivalence
ff(k,d,r)F (M, k) ' ff(k,d,r)F (N, k) ,
Remark. (1). Cohen and Taylor (unpublished manuscript) prove by very dif-
ferent methods that the configuration spaces of smooth manifolds are stable
homotopy invariant. In their work the bound on the number suspensions re-
quired to achieve homotopy invariance is significantly weaker.
Nevertheless, an advantage of their approach is its applicability to other
kinds of configuration spaces. For example, their results apply as well to the
unordered configuration spaces of a smooth manifold. We are unable to analyse
the latter using our methods.
We now single out two corollaries of our main result. Assume in what
follows that M is a connected closed PL manifold.
Corollary B. The suspension spectrum 1 F (M, k)+ is a homotopy invariant
of M.
The second corollary extends the work of Levitt [L ], who considered only
the case r = 2.
Corollary C (k = 2). If M is r-connected for some 0 r 2, then
2-rF (M, 2) is a homotopy invariant of M.
Conventions. We work in the category Top of compactly generated spaces. A
non-empty space is always (-1)-connected. A non-empty space is 0-connected
if it is path connected. It is r-connected for r > 0 if it is path connected and
its homotopy groups (with respect to a choice of basepoint) vanish in degrees
r. A map A ! B of spaces (with B non-empty) is r-connected if for any
choice of basepoint in B, the homotopy fiber with respect to this choice of
basepoint is an (r- 1)-connected space. A weak (homotopy) equivalence is an
1-connected map. If two spaces A and B are related by a chain of weak
equivalences, we will often indicate it by writing A ' B.
ON THE HOMOTOPY INVARIANCE OF CONFIGURATION SPACES 3
2.Fiberwise suspension
Let A ! X be a map of spaces. Define
TopA!X
to be the category of spaces "between A and X." Specifically, an object is a
space Y and a choice of factorization A ! Y ! X. A morphism is a map of
spaces which is compatible with their given factorizations. Call a morphism a
weak equivalence if it is a weak homotopy equivalence of underlying spaces.
We use the notation Top=X for Top;!X . If Y 2 Top=X is an object, define
its (unreduced) j-fold fiberwise suspension by
jXY := (Y x Dj) [ (X x Sj-1) ,
where the union is amalgamated over Y xSj-1. With respect to the first factor
projection map X x Sj- 1! X, we get a functor
jX:Top=X ! TopXx Sj-1!X .
Lemma 2.1. Let Y and Z be objects of Top=X whose underlying spaces are
path connected and have the homotopy type of CW complexes.
Assume for some j 0 that jXY and jXZ are weak equivalent objects.
Then there is a weak equivalence of spaces
jY ' jZ .
Proof. The statement is obviously true for j = 0, so we will assume that j > 0.
Moreover, we may assume that we are given a weak equivalence jXY !~ jXZ.
For any object T 2 Top=X , we have a cofibration sequence of spaces
X x Sj-1 ! jXT ! j(T+ ) ,
where we use the fact that j(T+ ) means T x Dj with T x Sj- 1collapsed to
a point. Using this cofiber sequence for both Y and Z, we get a commutative
diagram
jXY ---! j(Y+ )
? ?
' ?y ?y
jXZ ---! j(Z+ )
which is also homotopy pushout. It is well-known that cobase change pre-
serves weak equivalences (see e.g., Hirschhorn [H ]), so it follows that the map
j(Y+ ) ! j(Z+ ) is a weak equivalence.
Choose basepoints for Y and Z. Since j > 0, we have j(Y+ ) ' ( jY ) _ Sj
and similarly j(Z+ ) ' ( jZ) _ Sj. It follows that there is a weak equivalence
( jY ) _ Sj ' ( jZ) _ Sj .
4 MOKHTAR AOUINA AND JOHN R. KLEIN
Because Y and Z are connected, we have that jY and jZ are j-connected.
Using Lemma 2.2 below, we conclude that the composite
project j
jY include----!( jY ) _ Sj ' ( jZ) _-Sj---!( Z)
is a weak equivalence.
Lemma 2.2. Let U and V be j-connected spaces with j 0. Assume U and
V are equipped with non-degenerate basepoints. Assume h: U _ Sj ! V _ Sj
is a weak equivalence. Then the composition
project
g :U -include---!U _-Sjh--!V _ Sj ----! V
is also a weak equivalence.
Proof. Without loss in generality we can assume that U and V are CW com-
plexes with no cells in positive dimensions j. By cellular approximation, we
may also assume that h is a cellular map. Then h preserves j-skeleta, so there
is a commutative diagram
Sj - --! U _ Sj
? ?
h|Sj?y ?yh ,
Sj - --! V _ Sj
and it is straightforward to check that the left vertical map is a homotopy
equivalence. We infer that the map U ! V obtained by taking cofibers hori-
zontally is also a weak equivalence. But this map coincides with g.
3. Embeddings up to homotopy
Let K be a space. Write dim K k if K is up to homotopy a cell complex of
dimension k. We say that K is homotopy finite if it is homotopy equivalent
to a finite cell complex.
Let M be a PL manifold of dimension d, possibly with boundary. Fix a
map f :K ! M, in which K is a homotopy finite space.
Definition 3.1. An embedding up to homotopy of f is a pair
(N, h)
in which
o N denotes a compact codimension zero PL submanifold of the interior
of M, and
o h: K ! N is a homotopy equivalence such that composition
K !h N M
is homotopic to f.
ON THE HOMOTOPY INVARIANCE OF CONFIGURATION SPACES 5
A concordance of embeddings up to homotopy (N0, h0) and (N1, h1) of f
consists of
o an embedded PL h-cobordism
(W, N0, N1) (M x I, M x 0, M x 1) ,
where @W = N0[ @1W [ N1 and @1W is the internal part of @W , (i.e.,
W \ M x {i} = Ni for i = 0, 1);
o a homotopy equivalence
H :(Kx I, Kx 0, Kx 1) ~! (W, N0, N1)
which factors the map fx id up to homotopy.
We remark that our definition of embedding up to homotopy differs from the
one of Stallings and Wall in that we do not work with simple homotopy equiv-
alences. Our notion of concordance accounts for this distinction (Stallings and
Wall use s-cobordisms in their notion of concordance); our set of concordance
classes coincides with theirs when dim K d - 3.
Theorem 3.2 (Stallings [St], Wall [Wa1 ]). Assume dim K k d- 3. If
f :K ! M is (2k- d+ 1)-connected, then f embeds up to homotopy. Further-
more, any two embeddings up to homotopy of f are concordant whenever f is
(2k- d+ 2)-connected.
4. Decompression
Let (N, h) be an embedding up to homotopy of f :K ! M. If C denotes
the closure of the complement of N inside M, then C is an object of Top@M M .
Definition 4.1. The object
C 2 Top@M M
is called the complement of (N, h).
By considering the inclusion M x 0 M x Dj, and taking a compact
regular neighborhood of N in M x Dj, we have an associated embedding up
to homotopy of the composite
f j
fj: K ! M = M x 0 M x D .
Denote this embedding up to homotopy by (Nj, hj), where Nj ~=N x Dj and
the homotopy equivalence hj is identified the composite
K !h N N x Dj.
This new embedding up to homotopy is called the j-fold decompression of
(N, h). Its complement has the structure of an object of TopMx Sj-1 Mx Dj.
6 MOKHTAR AOUINA AND JOHN R. KLEIN
However, to avoid technical problems, we will henceforth regard the com-
plement as a space over M by projecting away from the Dj factor. That is,
we will think of the complement as an object of TopMx Sj-1!M .
Lemma 4.2 (Compare [Kl2 , x2.3]). Assume that M is closed. Then the com-
plement of (Nj, hj) is weak equivalent to the object
jMC .
Proof. The regular neighborhood Nj can be chosen as N x D1=2 M x Dj,
where D1=2 Dj is the disk of radius 1=2. The complement of (Nj, hj) is then
(M x Dj) - int(N x D1=2) = C x D1=2 [ M x D[1=2,1],
where D[1=2,1]denotes the annulus consisting of points in Dj whose norm varies
between 1=2 and 1. The above union is amalgamated over C x @D1=2.
The subspace of the complement given by (C x D1=2) [ (M x D1=2) is
evidently isomorphic to jMC. The inclusion map of this subspace is, up to
isomorphism, a morphism of TopMx Sj-1!M . Furthermore, this inclusion is a
weak homotopy equivalence of underlying spaces.
5. The suspended complement
Proposition 5.1. Assume f :Kk ! Md is an r-connected map, where M
is a closed connected PL manifold of dimension d, and k d- 3. Suppose
that f has two embeddings up to homotopy (N, h) and (N0, h0) with respective
complements C and C0. Then there is a homotopy equivalence,
jC ' jC0,
where j = max (2k - d - r + 2, 0).
Proof. By the Stallings-Wall theorem, with j = max (2k - d - r + 2, 0), we see
that the j-fold decompressions of (N, h) and (N0, h0) are concordant. Using
Lemma 4.2 we infer that there is a weak equivalence of objects
jMC ' jMC0.
By Lemma 2.1, we conclude jC ' jC0.
6. Proof of Theorem A
Suppose that M and N are homotopy equivalent r-connected (r 0) closed
PL manifolds of dimension d.
With appropriate modifications, we will argue along the lines of Levitt's
strategy for showing F (M, 2) ' F (N, 2) when M and N are 2-connected (see
[L ]).
ON THE HOMOTOPY INVARIANCE OF CONFIGURATION SPACES 7
Case 1: d 2. By the classification of low dimensional manifolds, M and N
are PL homeomorphic. It follows that F (M, k) and F (N, k) are homeomorphic
for all k.
Case 2: d > 2. Let
fatk(M) Mxk
denote the fat diagonal. This subpolyhedron is the space of k-tuples of points
of M such that at least two entries in the k-tuple coincide.
By choosing a regular neighborhood, we obtain an embedding up to homo-
topy of the inclusion fatk(M) Mxk . Its complement C is weak equivalent
to F (M, k) when the latter is considered as an object of Top=Mxk . Denote
this embedding up to homotopy by (V, h). Note that that have an associated
codimension one manifold splitting given by
(Mxk ; V, C; @V ) .
Repeat this procedure for the fat diagonal of N in Nxk to get an em-
bedding up to homotopy of the inclusion fatk(N) Nxk . Call the latter
embedding up to homotopy (W, h0). Its complement C0 is identified with
F (N, k) 2 Top=Nxk . Thus we have a codimension one splitting
(Nxk ; W, D; @W ) .
The next step is to choose a homotopy equivalence g :M !~ N. The k-fold
product of g with itself gives another homotopy equivalence gk: Mxk !~ Nxk .
We can therefore use gk to form another triad
(W [ C; W, C; @W )
together with a homotopy equivalence _ :Nxk ! W [ C (note that we are
using C instead of D). The latter triad is codimension one Poincar'e dual-
ity splitting. According to the Browder-Casson-Sullivan-Wall theorem [Wa2 ,
Th. 12.1], such splittings can be made into manifold splittings: there exists
codimension one manifold splitting (Nxk ; W 0, C0; @W 0) and a homotopy equiv-
alence of triads
OE: (Nxk ; W 0, C0; @W 0) ~! (W [ C; W, C; @W )
such that OE: Nxk ! W [ C is homotopic to _. These data describe another
embedding up to homotopy of the inclusion fatk(N) ! Nxk with the property
that its complement is identified with F (M, k) up to homotopy equivalence.
Summarizing thus far, we have two embeddings up to homotopy of the
inclusion fatk(N) ! Nxk , one whose complement is identified with F (N, k)
and the other whose complement is identified with F (M, k).
The next step of the argument is to verify the hypotheses of Proposition
5.1. One checks by elementary means that dim fatk(N) (k- 1)d. As d > 2,
the hypothesis (k- 1)d kd - 3 is satisfied. Furthermore, the inclusion map
8 MOKHTAR AOUINA AND JOHN R. KLEIN
fatk(N) ! Nxk is r-connected (recall that r is the connectivity of N). Hence,
applying 5.1, we infer
jD ' jC0,
where j = max (2(k- 1)d - kd - r + 2, 0). It is then straightforward to check
that j = ff(k, d, r).
Finally, we recall that D ' F (N, k) and C0 ' F (M, k). With respect to
these identifications, we get
ff(k,d,r)F (M, k) ' ff(k,d,r)F (N, k) .
This concludes the proof of Theorem A.
References
[B-C-T]Bödigheimer, C.-F., Cohen, F., Taylor, L.: On the homology of configura*
*tion
spaces. Topology 28, 111-123 (1989)
[H] Hirschhorn, P. S.: Model categories and their localizations. (Mathematic*
*al Surveys
and Monographs, Vol. 99). Amer. Math. Soc. 2003
[Kl2] Klein, J. R.: Poincar'e embeddings and fiberwise homotopy theory. Topolo*
*gy 38,
597-620 (1999)
[Kr] Kriz, I.: On the rational homotopy type of configuration spaces. Ann. of*
* Math.
139, 227-237 (1994)
[L] Levitt, N.: Spaces of arcs and configuration spaces of manifolds. Topol*
*ogy 34,
217-230 (1995)
[St] Stallings, J. R.: Embedding homotopy types into manifolds. 1965 unpubli*
*shed
paper (see http://math.berkeley.edu/~stall for a TeXed version)
[Wa1] Wall, C. T. C.: Classification problems in differential topology_IV. Thi*
*ckenings.
Topology 5, 73-94 (1966)
[Wa2] Wall, C. T. C.: Surgery on Compact Manifolds. (Mathematical Surveys and *
*Mono-
graphs, Vol. 69). Amer. Math. Soc. 1999
Mokhtar Aouina, Dept. of Mathematics, Wayne State University, Detroit, MI 48*
*202
E-mail address: aouina@math.wayne.edu
John R. Klein, Dept. of Mathematics, Wayne State University, Detroit, MI 482*
*02
E-mail address: klein@math.wayne.edu