Configuration spaces on the sphere and higher loop spaces
Paolo Salvatore
Abstract
We show that the homology over a field of the space n nX of free maps f*
*rom the n-sphere to
the n-fold suspension of X depends only on the cohomology algebra of X and *
*compute it explicitly.
We compute also the homology of the closely related labelled configuration *
*space C(Sn, X) on
the n-sphere with labels in X and of its completion, that depends only on t*
*he homology of X.
In many but not all cases the homology of C(Sn, X) coincides with the homol*
*ogy of n nX. In
particular we obtain the homology of the unordered configuration spaces on *
*a sphere.
MSC (2000): 55P48, 55R80, 55S12.
Keywords: Configuration space, loop space, homology operation.
Introduction
In order to compute the homology of the space of based maps n(Y ) = map*(Sn, Y*
* ) from the n-
sphere to a CW-complex Y , in general one needs to know a great deal of informa*
*tion on Y . However,
when Y = nX is a n-fold suspension, a classical result by Milgram [9] states t*
*hat the homology of
n nX depends just on the homology of X. An explicit description is given in [3*
*]. This depends upon
the existence of a small model for the mapping space. Namely the configuration *
*space C(Rn, X) of
pairwise distinct points in Rn with labels in X, modulo base point cancellation*
*, is homotopy equivalent
to n nX when X is connected [6]. In general n nX is the group completion of C*
*(Rn, X).
Let us turn our attention to the free mapping space nY = map(Sn, Y ). In the*
* case n = 1 the
homology of X depends only on the homology of X [8]. This depends again on th*
*e existence of
a small model for the mapping space, the configuration space C(S1, X) of points*
* in the circle with
labels in X [2], for X connected. This does not extend to all n. For X connecte*
*d the configuration
space C(Sn, X) is homotopy equivalent to the section space of a bundle over Sn *
*with fiber nX
[2], obtained from the tangent bundle of the sphere øn by adding a point at inf*
*inity to each fiber
and smashing it with X. In general this section space is a kind of completion o*
*f C(Sn, X) . When
n 6= 1, 3, 7, Sn is not parallelizable, so that one does not expect C(Sn, X) to*
* be homotopy equivalent
to the free mapping space n nX.
In this paper we compare the homotopy types of these spaces and we compute th*
*eir homology
with coefficients in any field. Up to homotopy there are fibrations C(Rn, X) ! *
*C(Sn, X) ! nX
(for X connected) and n nX ! n nX ! nX, induced by the evaluation at one poi*
*nt. Since the
basis of both fibrations is a suspension, we can reconstruct the total spaces b*
*y means of the clutching
functions n-1X x n nX ! n nX. It turns out that the clutching function of th*
*e first fibration
comes from the action of the little n-discs operad Sn-1 x X x C(Rn, X) ! C(Rn, *
*X). This action
adds to a `cloud' of labelled points in C(Rn, X) an extra point in the directio*
*n parametrized by Sn-1
with label parametrized by X. The same argument works when X is not connected. *
*In general we
get decompositions both for C(Sn, X) (Theorem 1) and its completion (Theorem 6)*
*, that coincide
for X connected. The decomposition is compatible with the Snaith splitting (Pro*
*position 5). The
clutching function induces in homology the so called Browder operation, that ha*
*s been computed
explicitly for C(Rn, X) in [3]. For example in characteristic 0 and for n = 2 w*
*e get the adjoint action
of the homology of X on the free Gerstenhaber algebra that it generates. This a*
*llows to determine
the homology of C(Sn, X), that depends only on the homology of X (Corollary 3).*
* In particular for
X = S0 we obtain the homology of the unordered configuration spaces on Sn. We w*
*rite down an
1
explicit basis of their homology in Theorem 18. The case when n is even and the*
* characteristic is
odd is not covered by the methods in [1].
Let us consider now the evaluation fibration of the mapping space n nX. In *
*this case the
clutching function is obtained by twisting the first argument Sn-1xX ! n nX an*
*d then applying
as before the little n-discs action Sn-1x( n nX)2 ! n nX (Theorem 11) . The tw*
*ist is the com-
position operation induced by the element c 2 ßn-1( n1Sn) adjoint to the Whiteh*
*ead product of the
generator ßn(Sn) with itself. This happens exactly because c classifies the fib*
*erwise compactification
of the tangent bundle of the n-sphere (Lemma 10).
Now c goes in homology (up to shift of component) to the Browder operation of*
* the identity
' 2 H0( nSn) with itself. This implies that the clutching function of the mappi*
*ng space in homology
depends only on the cohomology algebra of X (for finite type), and so does the *
*homology of n nX
(Corollary 12). We show in Example 16 that the homology of n nX does not depen*
*d just on the
additive homology of X. As an application we write down an explicit basis of th*
*e homology of the
mapping space nSm , for n m, in Theorems 18 and 19.
Let us compare the configuration space (or its completion for X not connected*
*) and the mapping
space: when c vanishes they are homotopy equivalent. This happens when either X*
* is a suspension,
or n = 1, 3, 7, or for n odd after inverting the prime 2 (Proposition 9). Moreo*
*ver the two spaces have
the same homology either mod 2 (Corollary 13), or for X connected and rationall*
*y (Corollary 14).
In general they do not have the same homology for n even at odd primes (Example*
* 16), or rationally
when X is not connected (Example 15).
We present also a homotopy pullback decomposition of our spaces, induced by d*
*ecomposing the
n-sphere of the domain as union of two discs (Propositions 20 and 21). In this *
*case the twist occurs
in the diagram describing the configuration space. We apply the induced sequen*
*ce of homotopy
groups to show in Example 22 that the configuration space and the mapping space*
* are not rationally
homotopic for n = 2 and X = CP2, but they have the same homology over all field*
*s.
It would be interesting to phrase our computations in terms of the higher Hoc*
*hschild homology
spectral sequence described in 5.9 of [10].
In sections 1,2 and 3 we consider respectively labelled configuration spaces *
*on the sphere, the
associated spaces of sections, and mapping spaces. In section 4 we compute som*
*e examples and
we compare the homology of the spaces above. In section 5 we present the homot*
*opy pullback
decomposition and an application.
I am grateful to Sadok Kallel for many helpful conversations.
1 Configuration spaces
Let X be a based CW-complex, not necessarily connected. Let n be a positive int*
*eger.
We recall some definitions from [2]. Let Fk(M) be the space of pairwise disti*
*nct k-tuples in a
manifold M. We consider the elements of the space Fk(M)x kXk as finite sets of *
*points`in M with a
label in X. The configuration space C(M, X) is the quotient of the disjoint uni*
*on k Fk(M) x kXk
under the equivalence relation induced by cancelling points labelled by the bas*
*e point of X. Given
a closed subset A M, the relative configuration space C(M, A; X) is the quoti*
*ent induced by
cancelling all points located in A and those labelled by the base point.
We identify Sn = Rn [ 1, based at infinity. Let Dn Rn be the unit disc. The*
* sequence Dn !
Sn ! (Sn, Dn), for X connected, induces a quasifibration C(Dn, X) ! C(Sn, X) ! *
*C(Sn, Dn; X)
(Lemma p.178 in [2]). Now nX C(Sn, Dn; X) is a strong deformation retract, *
*seen as the
subspace of configurations with a single labelled point [7]. We will construct *
*a homotopy pushout
decomposition of C(Sn, X), by pulling back to the total space_the_decomposition*
*_of nX = Sn ^ X
induced by splitting Sn as union of the n-discs Dn and En = Sn - Dn. The quasi*
*fibration over
Dn+ ^ X and En ^ X is homotopically trivial. It remains to glue the two pieces *
*together. This idea
works even when X is not connected.
2
We recall that C(Dn, X) ' C(Rn, X) is homotopy equivalent to the free algebra*
* on the operad
of little n-discs generated by X. In particular there is an action of the space*
* of two little n-discs
Dn(2) by a map æ2 : Dn(2) x C(Rn, X) x C(Rn, X) ! C(Rn, X). Let us restrict th*
*is map to
the deformation retract Sn-1 Dn(2) in the first factor and to X C(Rn, X), t*
*he subspace of
configurations consisting of a single labelled point located at the origin, in *
*the second factor. We get
a map ~ : Sn-1+^X xC(Rn, X) ! C(Rn, X), that we shall call the Browder map. Up *
*to deformation
~(z, x, c) is the configuration obtained from the configuration c by adding a p*
*oint outside c in direction
z with label x.
Theorem 1. There is a homotopy pushout
ß n
Sn-1+^ X x C(Rn, X)_-X x C(R , X)
| |
| |
~ || ||
| |
?| ?|
C(Rn, X)_________-C(Sn, X),
where ~ is the Browder map and ß the projection.
Proof.Let us consider the closed deformation retract K of C(Sn, X) containing t*
*he configurations
with at most one labelled point located outside Dn. The deformation is obtained*
* by pushing the
particles away from the north pole and deforming slightly the labels towards th*
*e base point. Let
A, B K be the closed subsets such that the point located outside Dn, if it ex*
*ists, is respectively in
2Dn and 2En. Clearly A[B = K. Moreover the inclusion C(Dn, X) A is a homotopy*
* equivalence.
On the other hand A \ B ~=Sn-1+^ X x C(Dn, X), and B ~=Dn+ ^ X x C(Dn, X). With*
* these
identifications the map A \ B ! B is induced by the inclusion Sn-1 Dn. Moreov*
*er the inclusion
A \ B ! A is homotopic to the Browder map. __
|__|
From now on we consider homology with coefficients in a field K. We recall th*
*e definition of the
Browder operation: let Y be a space acted on by the little n-discs operad. Let *
*en-12 Hn-1(Dn(2))
be the class represented by the deformation retract Sn-1 Dn(2). Then the Brow*
*der operation, for
x 2 Hp(Y ) and y 2 Hq(Y ), is defined by [x, y] := æ2*(en-1 x y) 2 Hp+q+n-1(*
*Y ). This differs
from (5.7) of [3] by the sign (-1)(n-1)p+1.
For n = 1 the fundamental class of S0 is {1} - {-1}, so that the Browder oper*
*ation is the
commutator of the Pontrjagin product.
Definition 2. The Browder action is the homomorphism
[ , ] = ~* : n-1~H(X) H(C(Rn, X)) ! H(C(Rn, X)).
Recall from Theorem 3.1 in [3] that the homology H~(C(Rn, X)) depends functor*
*ially on the
homology ~H(X). For example in characteristic 0 and for n > 1 the former is the*
* free n-algebra on
the latter. A n-algebra is a graded variation of a Poisson algebra [11]. In our*
* case the Pontrjagin
product is the commutative product and the Browder operation is the bracket (up*
* to sign). The
Browder action for char(K) = 0 is exactly the adjoint action of the space of ge*
*nerators ~H(X) on
H(C(Rn, X)).
In positive characteristic there are additional homology operations generatin*
*g H(C(Rn, X)) over
H(X), the unstable Dyer-Lashof operations. See (1.1),(1.3) of [3] and Propositi*
*on 17.
In general the Browder action can be computed explicitly and depends just on *
*the homology of
X by Thm. 1.2 (5,6,8) and Thm. 1.3 (4) of [3]. For example mod 2 the action *
*is trivial on all
Dyer-Lashof operations except the top one. Recall that H(C(R, X)) is the free a*
*ssociative algebra
on ~H(X).
The Mayer-Vietoris sequence implies the following corollary.
3
Corollary 3. The homology of C(Sn, X) depends only on the homology of X and is *
*the direct sum
of the suspended kernel and the cokernel of the Browder action on the homology *
*of C(Rn, X).
The configuration spaces C(M, X), for a manifold M, are naturally filtered by*
* subspaces Ck(M, X),
having at most k particles. We denote the subquotients by Dk(M, X) = Ck(M, X)=C*
*k-1(M, X) and
Dnk(X) = Dk(Rn, X). For example the space Dk(Sn, S0) is the unordered configura*
*tion space on the
sphere Ck(Sn) = Fk(Sn)= k.
The homology H(C(M, X)) splits naturally as sum of H(Dk(M, X)), as k varies. *
*Compare Lemma
4.2 p.238 in [3].LThis explains the first assertion of Corollary 3, as
H(C(Sn, X)) ~= kH(Fk(Sn); ~H(X) k).
The Browder map is compatible with the filtration by number of particles, so *
*that we have the
following refinement:
Corollary 4. The reduced homology of Dk(Sn, X), for k > 1, is the direct sum of*
* the suspended
kernel and the cokernel of the Browder action n-1~H(X) ~H(Dnk-1(X)) ! ~H(Dnk*
*(X)).
There is a geometric version of this corollary. Recall that C(M, X) is stably*
* homotopy equivalent
to the wedge sum of Dk(M, X), indexed over k [2]. The Browder map defines a st*
*able map bk :
n-1X ^ Dk-1(X) ! Dk(X), by restriction to the relevant wedge summands. For exa*
*mple the map
b2 is the restriction to the stable summand Sn-1 ^ X ^ X of the quotient map Sn*
*-1+^ X ^ X !
Sn-1+^Z2(X ^ X).
Proposition 5. The k-th summand of the stable splitting Dk(Sn, X) is stably hom*
*otopy equivalent
to the cofiber of bk : n-1X ^ Dnk-1(X) ! Dnk(X) for k > 1 and to nX _ X for k*
* = 1.
Proof.We obtain a homotopy pushout (k) expressing the homotopy type of Ck(Sn, X*
*), if we replace,
in the diagram of Theorem 1, C by Ck-1 in the top row and by Ck in the bottom r*
*ow. For k = 1
this proves the proposition. For k > 1 the cofiber diagram of (k - 1) and (k) g*
*ives Dk(Sn, X) as
homotopy pushout. The top map ßk of such diagram is a stable retraction, and bk*
* is exactly the__
restriction of the left hand side map ~k to the stable kernel of ßk. *
* |__|
2 Section spaces
We recall that there is a `scanning' map C(Dn, X) ! n nX ~=map*(Dn=Sn-1, Dn=Sn*
*-1 ^ X).
Let i : X ! n nX be the adjoint of the identity. Roughly speaking, if we scan *
*a configuration, we
obtain a n-fold loop, sending small discs centered at the configuration points *
*to nX ~=Dn=Sn-1^X,
by identifying them to the unit disc and applying i to the label of the center.*
* The n-fold loop sends
everything else to the base point.
The scanning map can be extended to C(Sn, X) by using the exponential map of *
*the sphere. Let
nX ! ø+n(X) ! Sn be the bundle obtained by smashing X fiberwise with the fiber*
*wise one-point
compactification ø+nof the tangent bundle øn of the n-sphere. Then the scanning*
* map goes from
C(Sn, X) to the space (ø+n(X)) of sections of the bundle. Up to homotopy the s*
*canning maps fit
into a diagram from the sequence
C(Dn, X) ! C(Sn, X) ! C(Sn, Dn; X)
to the evaluation fibration ev
n nX ! (ø+n(X)) ___- nX.
The scanning maps C(Dn, X) ! n nX and C(Sn, X) ! (ø+n(X)) are homotopy equi*
*valences
if and only if X is connected.
We can define the Browder map for the n-fold loop space n nX, similarly as f*
*or the configuration
space, by using the action Dn(2)x n nX x n nX ! n nX of the n-discs operad, an*
*d restricting
the action in the second factor to the subspace X ___i- n nX.
4
If X is connected then the next theorem reduces to Theorem 1. Also the proof *
*is an adaptation
of the proof of Theorem 1 to the general case.
Theorem 6. There is a homotopy pushout
ß n n
Sn-1+^ X x n nX __-X x X
| |
| |
~|| ||
| |
?| ?|
n nX ________- (ø+n(X)),
where ~ is the Browder map and ß the projection.
Proof.Let ev : (ø+n(X)) ! øn(N)+ ^ X ~= nX be the fibration evaluating at the *
*north pole N.
We construct a map (ff, fi, fl) from the diagram
Dn+ ^ X x n nX oe__ Sn-1+^ X x n nX ___- M(~),
where M(~) is the mapping cylinder of the Browder map, to the diagram
ev-1(Dn+ ^ X) oe__ ev-1(Sn-1+^ X) ___- ev-1(En ^ X).
We construct ff so that it covers an automorphism of Dn+ ^ X homotopic to -1 *
*^ X. We identify
both the southern and the northern hemisphere to the unit disc, by projection f*
*rom the opposite
pole. This gives a trivialization of the tangent bundle on each hemisphere.
The map ff(z ^ x, y) 2 (ø+n(X)) is defined by y 2 Map(Dn, @Dn; nX) on the s*
*outhern hemi-
sphere. Let OE : (Dn, Sn-1) ! (Sn, 1) be the relative homeomorphism OE(x) = x=*
*(1 - |x|). In
the northern hemisphere we have the adjoint n-fold loop ff(z ^ x, y)(w) = OE(2(*
*w - z=4)) ^ x if
|w - z=4| 1=2, and otherwise w goes to the point at infinity.
Similarly, on the cylinder I x Sn-1+^ X x n nX M(~), fl(t, z ^ x, y) is de*
*fined in the southern
hemisphere by y. On the northern hemisphere fl(t, z ^ x, y)(w) = OE(2(w - z(t *
*+ 1)=4)) ^ x for
|w - z(t + 1)=4| 1=2 and otherwise w goes to the point at infinity. The map f*
*l sends the end of the
mapping cylinder n nX to ev-1(*) by projection from the north pole.
Both ff and fi are homotopy equivalences, because they cover an automorphism *
*of the basis and
induce homotopy equivalences of the fibers over points of all components. Final*
*ly fl is a homotopy
equivalence, since the domain has n nX as deformation retract, identified by f*
*l to ev-1(*) '
ev-1(En ^ X). By the homotopy invariance of homotopy pushouts the theorem is_pr*
*oved._
|__|
Also in this case we call Browder action the homomorphism ~* : n-1~H(X) H(*
* n nX) !
H( n nX) induced by the Browder map.
Corollary 7. The homology of (ø+n(X)) depends only on the homology of X, and i*
*s the direct sum
of the suspended kernel and the cokernel of the Browder action on the homology *
*of n nX.
Proof.Recall that H( n nX) is obtained from H(C(Rn, X)), according to the group*
* completion
theorem, by adding the inverses of the components.
In the case n > 1 ß0(C(Rn, X)) = N[ß0(X)], and ß0( n nX) = Z[ß0(X)]. We must *
*only compute
the Browder action on the inverse of a component a 2 ß0(X) H( n nX). But by t*
*he Poisson
relation and since the Browder action on the unit is trivial, we obtain [x, a-1*
*] = -[x, a]a-2.
In the case n = 1 ß0(C(R, X)) and ß0( X) are respectively the free monoid an*
*d the free group
on the based set ß0(X), and the Pontrjagin product is clear. __
|__|
Example 8. If X = S0_S0, then (ø+n(X)) is homotopy equivalent to the section s*
*pace of the bundle
F3(Rn+1) ! F2(Rn+1) defined by forgetting the third point of a configuration [5*
*].
5
3 Higher loop spaces
In some special cases the section space and the mapping space coincide.
Proposition 9. The fibration nX ! ø+n(X) ! Sn is trivial, so that the section *
*space is homotopic
to the ordinary mapping space n nX, in the following cases:
1.When X is a suspension X = Y ;
2.For n = 1, 3, 7;
3.For n odd and away from the prime 2.
Proof.1) The tangent bundle of the sphere is trivialized by adding a trivial li*
*ne bundle. Thus
ø+n( Y ) ~=(øn R)+(Y ) ~= n+1Y x Sn.
2) The spheres S1, S3, S7 are parallelizable.
3) Follows from the following lemma, since 2['n, 'n] = 0 2 ß2n-1(Sn) for n od*
*d, where 'n_2_ßn(Sn)
is the generator and the brackets denote the Whitehead product. *
* |__|
Lemma 10. The fibration Sn ! ø+n! Sn with section Sn ! ø+nat infinity is classi*
*fied by the
adjoint of the Whitehead product ['n, 'n] in ßn-1( n1Sn).
Proof.The bundle ø+nis trivial if we forget the section, because it is the sphe*
*re bundle of øn R,
and its section can be identified to the diagonal : Sn ! Sn x Sn.
We will show that the homotopy class of the clutching function c : Sn-1 ! n1*
*Sn of the fibration
ø+nis exactly the boundary of 'n 2 ßn(Sn) in the long exact sequence associated*
* to the evaluation
fibration n1Sn ! n1Sn ! Sn. The index 1 denotes the component of degree 1 map*
*s. We identify
Sn-1 to the equator of Sn. We take the north pole N and the south pole S respec*
*tively as base
points in the domain and the range.
Let H : I x Sn-1 ! SO(n + 1) ! n1Sn be the transformation such that H(t, a) *
*rotates by
tß the plane generated by a and N, and fixes the orthogonal complement. The clu*
*tching function
of ø+nis c = H(1, _) : Sn-1 ! n1Sn. Let us identify the unreduced suspension *
*of Sn-1 to Sn
via [t, a] 7! H(t, a)(N). Then H defines an element in ßn( n1Sn, n1Sn) project*
*ing to the generator
'n 2 ßn(Sn), and c = @'n. But by a theorem of Whitehead [12] @('n) is adjoint t*
*o_['n, 'n], up to sign
convention. |*
*__|
Let w : Sn-1+^ X ! n nX be the composite of c+ ^ i : Sn-1+^ X ! n1Sn+^ n n*
*X and of the
composition operation o : n1Sn+^ n nX ! n nX given by o(_, f) = f_.
We call the map k(s, x, y) = ~(s, w(s, x), y) the twisted Browder map.
Theorem 11. There is a homotopy pushout
ß n n
Sn-1+^ X x n nX __-X x X
| |
| |
k|| ||
| |
?| ?|
n nX _________- n nX,
where k is the twisted Browder map and ß is the projection.
Proof.We identify the two hemispheres to the unit disc by orientation preservin*
*g diffeomorphisms
and we proceed as in Theorem 6. The main difference is that the trivializations*
* regard the domain
but not the range of the mapping space.
Thus the twisted Browder map k replaces ~ in the proof, by comparing the triv*
*ializations of the
one-point compactified tangent bundle on the two hemispheres with Lemma 10. __
|__|
6
In the next corollary X is supposed to have finite type. Otherwise we must re*
*place the expression
`cohomology algebra' by `homology coalgebra'.
Corollary 12. The homology of n nX with coefficients in a field depends only o*
*n the cohomology
algebra of X and is the direct sum of the suspended kernel and the cokernel of *
*the twisted Browder
action .
Proof.Let ' 2 H0( n1Sn) be the class of the identity map. By the diagram at p.*
* 215 in [3] the
Hurewicz homomorphism sends the adjoint Whitehead product in ßn-1( n2Sn) to the*
* Browder op-
eration [', '] 2 Hn-1( n2Sn), with our sign convention. Therefore, for x 2 H(X)*
*, w*(en-1 x) =
o*([', ']'-1 x) is the composition product in homology, wherePwe identify x t*
*o i*(x) 2 H( n nX).
Suppose that the iterated coproduct : H(X) ! H(X3) is (x) = ix0i x00i x0*
*00i.
Let Ø be the conjugation of the Hopf algebra H( n nX). It depends only on the*
* cohomology
algebra of X, by induction and by Proposition 1.5 of [3]. P
Then the twisted Browder action is k*(x y) = [x, y] + i[x0i, x00i]Ø(x000i)*
*y.
Compare also Theorem 3.2 (iv) of [4]. __
|__|
4 Examples and applications
Corollary 13. At the prime 2 the homology of n nX depends just on the homology*
* of X, and is
isomorphic to the homology of the section space (ø+n(X)).
Proof.The Browder operation [', '] is trivial mod 2. *
* |___|
Corollary 14. If X is connected then n nX and (ø+n(X)) have the same rational*
* homology.
Proof.Rationally any suspension is a wedge of spheres, so that nX is rationall*
*y a (n + 1)-fold
suspension n+1Y , but n n+1Y ' (ø+n( Y )) has the same homology as (ø+n(X))*
*_by Corollary
7. |__|
The corollary is not true in general, as the following example shows.
Example 15. The spaces 2(S2 _ S2) and (ø+2(S0 _ S0)) have not the same ration*
*al homology
componentwise.
Proof.Both spaces have components indexed by Z x Z. The bifiltration of the sec*
*tion space =
(ø+2(S0 _ S0)) by fiberwise bidegree is compatible with the bifiltration, by n*
*umber of particles, of
the configuration space C(S2, S0_ S0) of bicoloured particles on the sphere. Co*
*mpare also Example
8. The bifiltration of the mapping space M = 2(S2 _ S2) is given by ordinary b*
*idegree.
Let us denote the generators of ~H0(S0 _ S0) ~H0( 2(S2 _ S2)) by x and y. I*
*t turns out from
Corollary 7 that the homology group H1( m,n) is the quotient of
H1( 2m,n(S2 _ S2)) = Q{[x, x]xm-2yn, [x, y]xm-1yn-1, [y, y]xm yn-2}
by the subspace generated by [x, xm-1yn] and [y, xm yn-1]. But these elements *
*have respectively
coordinates (m-1, n, 0) and (0, m, n-1) with respect to the basis above. It fol*
*lows that H1( i,j) ~=Q2
for (i, j) = (1, 1) or (i, j) = (0, 1) or (i, j) = (1, 0) and H1( i,j) ~=Q in a*
*ll other cases.
On the other hand by Corollary 12 H1(Mm,n) is the quotient of H1( 2m,n(Sn _ S*
*n)) by the sub-
space generated by [x, xm-1yn] + [x, x]xm-2yn and [y, xm yn-1] + [y, y]xm yn-2,*
* that have coordinates
(m, n, 0) and (0, m, n). Thus H1(M0,0) ~=Q3 and H1(Mi,j) ~=Q for any (i, j) 6= *
*(0, 0). In particular
M0,0differs in homology by all components of . __
|__|
The next example shows that for n even and at odd primes the homology of the *
*mapping space
n nX does not depend just on the additive homology of X, so that in particular*
* the section space
and the mapping space have not the same homology.
7
Example 16. The spaces X = S2 _ S4 _ S6 and Y = CP3 have the same additive homo*
*logy , but
H9( 2 2X; Z3) ~=Z3 Z3 and H9( 2 2Y ; Z3) ~=Z3.
Proof.Let use denote the homology generators of the homology of X = S2 _ S4 _ S*
*6 by a1, a2, a3.
We must compute the Browder action mod 3
M
~i: H~j(X) Hi-1-j( 2 2X) ! Hi( 2 2X)
j
for i = 8, 9. For i = 8 the left hand side is generated by a1 [a1, a1] so th*
*at ~8 = 0 by The-
orem 1.2 (6) in [3]. For i = 9 the left hand side has dimension 6, with genera*
*tors a1 a3, a1
a31, a1 a1a2, a2 a2, a2 a21, a3 a1, and the right hand side has dimensi*
*on 5, with generators
[a1, a3], [a1, a1]a2, [a1, a2]a1, [a2, a2], [a1, a1]a21.
By the Poisson relation (Theorem 1.2 (5) in [3]) [a1, a31] = 3[a1, a1]a21= 0 *
*mod 3. Thus the
image of ~9 has dimension 4, with generators [a1, a3], [a1, a1]a2+ [a1, a2]a1, *
*[a2, a2], 2[a2, a1]a1, and
H9( 2 2(S2 _ S4 _ S6); Z3) ~=ker(~8) coker(~9) ~=Z3 Z3.
Let us consider now the case of the projective space CP3, with homology gener*
*ators e1, e2, e3. We
must check the twisted Browder action
M
ki: H~j(CP 3) Hi-1-j( 2 2CP3) ! Hi( 2 2CP3)
j
for i = 8, 9. By the proof of Corollary 12 k*( e1 y) = [e1, y], k*( e2 y) = [*
*e2, y] + [e1, e1]y and
k*( e3 y) = [e3, y] + 2[e1, e2]y - [e1, e1]e1y.
For the same reason as above k8 = 0. Moreover the image of k9 is generated by t*
*he five elements
[e1, e3], [e1, e1]e2+ [e1, e2]e1, [e1, e1]e2+ [e2, e2], 2[e1, e2]e1+ [e1, e1]e2*
*1, [e1, e3] + 2[e1, e2]e1- [e1, e1]e21.
Elementary linear algebra shows that they are linearly independent. Thus k9 i*
*s surjective, and
H9( 2 2CP3; Z3) ~=ker(k8) coker(k9) ~=Z3. __
|__|
We compute next the homology of nSn+k for k 0. We recall the computation o*
*f the homology
of nSn+k for n > 1.
Let X be a space acted on by the operad of little n-discs. For example X = n*
*Y or X = C(Rn, Y ).
We use the lower index notation for Dyer-Lashof operations and do not consider *
*Q0 here.
a) Case p = 2:
There are operations Qi: Hq(X; Z2) ! H2q+i(X; Z2) for 0 < i < n. (These opera*
*tion are also
named ,iin [3]). A string QI of symbols Qirepresents the composite operation. I*
*t is admissible if the
sequence of indices is weakly monotone. The empty string is admissible and repr*
*esents the identity.
b) Case p odd:
There are operations Qi(x) : Hq(X; Zp) ! Hpq+i(p-1)(X; Zp), defined for 0 < i*
* < n and when i
and q have the same parity. Note that the operations Qikeep the parity and the *
*Bockstein operator
fi switches parity. An ordered string QI of symbols Qi and fi represents the co*
*mposite operation.
We say that the operation QI(x) is admissible if
1) the last symbol of the non-empty string QI is Qi, and i has the same parit*
*y as the degree of x;
2) the sequence of indices is weakly monotone;
3) the indices of two adjacent Q0s have the same parity;
4) the indices of two Q0s separated by a fi have opposite parity;
5) two adjacent fi do not appear.
In particular the empty string is admissible and represents the identity.
c) In characteristic 0 only the identity is admissible.
8
Recall that C(Rn, Sk) ' nSn+k for k > 0 [2]. Let ' 2 Hk(C(Rn, Sk)) be the fu*
*ndamental class
of Sk ' C1(Rn, Sk).
Proposition 17. [3]
1) The homology of C(Rn, Sk) is the free commutative graded algebra on the ad*
*missible operations
QI('), and in addition on the admissible operations QJ([', ']) if n and k have *
*the same parity and
char(K) 6= 2.
2) The homology of nSn is obtained from C(Rn) = C(Rn, S0) by inverting forma*
*lly '.
3) The basis of H(C(Rn)) consisting of products of admissible operations has *
*an integral degree
that keeps track of the component, corresponding to the number of particles. Th*
*is degree is generated
by the rules (1) = 0, (') = 1, ([', ']) = 2, (xy) = (x) + (y) and (Qi(x)*
*) = (fiQi(x)) = p (x).
Now we are ready to state the computation. The next theorem for k > 0 gives a*
*lso the homology
of nSn+k ' C(Sn, Sk) by Proposition 9(1). Let F denote the graded commutative *
*algebra generated
by a set of vectors, and let Q be the set of all admissible operations on ' and*
* [', '] except the identity.
Theorem 18.
1) For n + k odd or for char(K) = 2
H(C(Sn, Sk)) ~=H(C(Rn, Sk)) n+kH(C(Rn, Sk)).
2) For n odd, k odd and char(K) 6= 2
H( nSn+k) ~=H( nSn+k)=([', ']) n+kH( nSn+k)=(').
3) For n even , k even and p = char(K) odd
H(C(Sn, Sk)) ~=F(', Q) 'p-1[', ']F('p, Q) n+k[', ']F(', Q) n+kF*
*('p, Q).
4) For n even, k even and char(K) = 0
H(C(Sn, Sk)) ~=F(') n+k[', ']F(') n+k{1}.
5) In the case k = 0 the component degree of C(Sn) is obtained from the degre*
*e of C(Rn) by the
additional rule ( n+kx) = (x) + 1.
Proof.We apply Corollary 7. The Browder action is given by [', _]. In case 1) t*
*he bracket [', '] is
trivial. In all cases the bracket [', x] is trivial if x is any generator of th*
*e algebra H( nSn+k) other
than '. Namely [', [', ']] = 0 by 1.2(6) of [3], and consequently [', QI(')] = *
*0 and [', QJ([', '])] = 0 by
1.2(8) and 1.3(9) of [3]. The action is a derivation with respect to the produc*
*t by 1.2(5) of [3]. This
completes the computation. __
|__|
The computation of the homology of configuration spaces of even spheres at od*
*d primes is not
deducible from the methods of [1].
It remains to compute the homology of nSn.
Theorem 19.
1) For n odd or char(K) = 2 H( nSn) = H( nSn) H(Sn).
2) For n even and char(K) = 0 H( n0Sn) = H(Sn-1) H(Sn) and H( nkSn) ~=H(S2n-*
*1) for
k 6= 0.
3) For n even and char(K) = p odd H( nkSn) ~=H( n0Sn) H(Sn) if p divides k and
H( nkSn) ~=F(Q) n[', ']F(Q) if (p, k) = 1.
9
Proof.Since the bundle ø+nis trivial, there is a homeomorphism (ø+n) ~= nSn, a*
*nd we can use
either Corollary 7 or Corollary 12. For n odd both the untwisted and the twiste*
*d Browder actions
are trivial. For n even note that the base point of (ø+n), the section at infi*
*nity, is the antipodal
map of Sn, that has degree -1. If k is the component of (ø+n) of fiberwise de*
*gree k, then k+1~=
nkSn. This is compatible with corollaries 7 and 12: the Browder action x 7! ['*
*, x] from nkSn to
nk+1Sn computes the homology of k+1, but the twisted Browder action from nk-*
*1Sn to nkSn
ø(x) = [', x] + [', ']'-1x computes the homology of nkSn. They are compatible*
* because 'ø(x) =
[', x]' + [', ']x = [', 'x]. Cases 1) and 2) are easy. In case 3) the multipl*
*ication by 'p induces an
isomorphism H( nkSn) ~=H( nk+pSn). If (p, k) = 1 then the composition by a degr*
*ee k map induces
an isomorphism H( n1Sn) ~=H( nkSn). One concludes the computation by keeping t*
*rack of the
component degree. __
|__|
5 Homotopy pullbacks
Another approach to our spaces is given by homotopy pullbacks, by decomposing t*
*he sphere Sn, basis
of the evaluation fibration, as union of two n-discs.
Proposition 20. The mapping space is the homotopy pullback
n nX _____- nX
| |
| |
| c|
| |
| |
?| c ?|
nX ____- n-1 nX,
where c is adjoint to the projection Sn-1+^ nX ! nX.
Proposition 21. The section space (ø+n(X)) is the homotopy pullback
(ø+n(X))____- nX
| |
| |
| c |
| |
| |
?| v ?|
nX ____- n-1 nX,
where v : nX ! n-1 nX is adjoint to w : Sn-1+^ X ! n nX.
The section and mapping spaces are both total spaces of fibrations with same *
*fiber and basis. The
mapping space evaluation fibration admits the section given by constant loops. *
*Therefore for i > 0
ßi( n nX) = ßi+n( nX) ßi( nX). The section space fibration admits just a stab*
*le section, given
by the inclusion Sn+^ X = C1(Sn, X) ! (ø+n(X)), since (Sn+^ X) ' ( nX _ X).
Example 22. C(S2, CP2) and 2 2CP2 are not rationally homotopy equivalent, but *
*they have the
same homology over any field.
Proof.We show first that the ß05s of the two spaces have different ranks. Ratio*
*nally 2CP2 ' S4_S6,
so that rk(ß5( 2 2CP2)) = rk(ß7(S4 _ S6)) + rk(ß5(S4 _ S6)) = 1.
The fiber square of Proposition 21 induces a long sequence of rational homoto*
*py groups
. .ß.i(C(S2, CP2)) ___- ßi( 2CP2) ßi( 2CP2) _____c#-v#ßi( 2CP2) ___- ßi-1(C*
*(S2, CP2)) . . .
Now ß5(S4 _ S6) has rank 0, so that rationally ß5(C(S2, CP2)) is the quotient*
* of ß6( 2CP2) =
Q{s, u} by the image of c# v#, defined on ß6( 2CP2)2 = Q{s1, s2}.
10
Here s1 and s2 represent the generators coming from the wedge summand S6 ! S4*
* _ S6 and
s = c#(s1) = c#(s2). Moreover u is the composition S6 ! S4 ! S4 ! (S4 _ S6),*
* with the first
map adjoint to the Whitehead product ['4, '4] : S7 ! S4.
The group [S1+^ CP2, 2 2CP2] splits as [ CP 2, 2 2CP2] [CP 2, 2 2CP2], b*
*ecause after
two suspensions S1+and S1 _ S0 are equivalent as co-H spaces. Rationally the f*
*irst summand
ß5( 2 2CP2) = Q{~u} has rank 1 and is detected by the action on ß5 ~=H5.
But w : S1+^ CP2 ! 2 2CP2 induces the nontrivial action w*([S1] e2) = [e1,*
* e1] where
ei2 H2i(CP 2) is the generator, by the proof of Corollary 12. Thus w correspond*
*s in the decomposition
above to ~u i, with i adjoint to the identity. By taking adjoints, v : 2CP2 !*
* 2CP2 represents
the sum in the group [ 2CP2, 2CP2] of the injection c : 2CP2 ! 2CP2 and th*
*e composition
2CP2 ! S6 ___u- 2CP2.
It follows that v#(s2) = s + u, and ß5(C(S2, CP2)) has rank 0.
The second assertion follows from Corollary 13 and the fact that away from th*
*e prime 2 2CP2 '
S4 _ S6. __
|__|
This shows that the rational homotopy type of C(S2, X) does not depend just o*
*n the rational
homology of X, by taking X = CP2and X = S2 _ S4.
Dipartimento di Matematica,
Universit`a di Roma öT r Vergata",
Via della Ricerca Scientifica 1,
00133 Roma, Italy
e-mail: salvator@mat.uniroma2.it
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11