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stable splitting of mapping spaces

Contents

Contents

Idea

The stabilization/suspension spectrum Σ Maps(X,A)\Sigma^\infty Maps(X,A) of mapping spaces Maps(X,A)Maps(X,A) between suitable CW-complexes X,AX, A happens to decompose as a direct sum of spectra (a wedge sum) in a useful way, related to the expression of the Goodwillie derivatives of the functor Maps(X,)Maps(X,-) and often expressible in terms of the configuration spaces of XX.

Definition

The stable splitting of mapping spaces discussed below have summands given by configuration spaces of points, or generalizations thereof. To be self-contained, we recall the relevant definitions here.

The following Def. is not the most general definition of configuration spaces of points that one may consider in this context, instead it is streamlined to certain applications. See Remark below for comparison of notation used here to notation used elsewhere.

Definition

(configuration spaces of points)

Let XX be a manifold, possibly with boundary.

For nn \in \mathbb{N}, the configuration space of nn distinguishable points in XX disappearing at the boundary is the topological space

(1)Conf n ord(X)(X nΔ X n)/(X n) \mathrm{Conf}^{ord}_{n}(X) \;\coloneqq\; \big( X^n \setminus \mathbf{\Delta}_X^n \big) / \partial(X^n)

which is the complement of the fat diagonal Δ X n{(x i)X n|i,j(x i=x j)}\mathbf{\Delta}_X^n \coloneqq \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \} inside the nn-fold product space of XX with itself, followed by collapsing any configurations with elements on the boundary of XX to a common base point.

Then the configuration space of nn in-distinguishable points in XX is the further quotient topological space

(2)Conf n(X)Conf n ord(X)/Σ n=((X nΔ X n)/(X n))/Σ(n), \mathrm{Conf}_{n}(X) \;\coloneqq\; Conf_n^{ord}(X)/\Sigma_n \;=\; \Big( \big( X^n \setminus \mathbf{\Delta}_X^n \big) / \partial(X^n) \Big) /\Sigma(n) \,,

where Σ(n)\Sigma(n) denotes the evident action of the symmetric group by permutation of factors of XX inside X nX^n.

More generally, let YY be another manifold, possibly with boundary. For nn \in \mathbb{N}, the configuration space of nn points in X×YX \times Y vanishing at the boundary and distinct as points in XX is the topological space

(3)Conf n(X,Y)(((X nΔ X n)×Y n)/(X n×Y n))/Σ(n) \mathrm{Conf}_{n}(X,Y) \;\coloneqq\; \Big( \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^n \big) / \partial(X^n \times Y^n) \Big) /\Sigma(n)

where now Σ(n)\Sigma(n) denotes the evident action of the symmetric group by permutation of factors of X×YX \times Y inside X n×Y n(X×Y) nX^n \times Y^n \simeq (X \times Y)^n.

This more general definition reduces to the previous case for Y=* 0Y = \ast \coloneqq \mathbb{R}^0 being the point:

Conf n(X)=Conf n(X,*). \mathrm{Conf}_n(X) \;=\; \mathrm{Conf}_n(X,\ast) \,.

Finally the configuration space of an arbitrary number of points in X×YX \times Y vanishing at the boundary and distinct already as points of XX is the quotient topological space of the disjoint union space

Conf(X,Y)(n𝕟((X nΔ X n)×Y k)/Σ(n))/ Conf\left( X, Y\right) \;\coloneqq\; \left( \underset{n \in \mathbb{n}}{\sqcup} \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^k \big) /\Sigma(n) \right)/\sim

by the equivalence relation \sim given by

((x 1,y 1),,(x n1,y n1),(x n,y n))((x 1,y 1),,(x n1,y n1))(x n,y n)(X×Y). \big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n) \big) \;\sim\; \big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}) \big) \;\;\;\; \Leftrightarrow \;\;\;\; (x_n, y_n) \in \partial (X \times Y) \,.

This is naturally a filtered topological space with filter stages

Conf n(X,Y)(k{1,,n}((X kΔ X k)×Y k)/Σ(k))/. Conf_{\leq n}\left( X, Y\right) \;\coloneqq\; \left( \underset{k \in \{1, \cdots, n\}}{\sqcup} \big( ( X^k \setminus \mathbf{\Delta}_X^k ) \times Y^k \big) /\Sigma(k) \right)/\sim \,.

The corresponding quotient topological spaces of the filter stages reproduces the above configuration spaces of a fixed number of points:

Conf n(X,Y)Conf n(X,Y)/Conf (n1)(X,Y). Conf_n(X,Y) \;\simeq\; Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y) \,.
Remark

(comparison to notation in the literature)

The above Def. is less general but possibly more suggestive than what is considered for instance in Bödigheimer 87. Concretely, we have the following translations of notation:

here: Segal 73, Snaith 74: Bödigheimer 87: Conf( d,Y) = C d(Y/Y) = C( d,;Y) Conf n( d) = F nC d(S 0)/F n1C d(S 0) = D n( d,;S 0) Conf n( d,Y) = F nC d(Y/Y)/F n1C d(Y/Y) = D n( d,;Y/Y) Conf n(X) = D n(X,X;S 0) Conf n(X,Y) = D n(X,X;Y/Y) \array{ \text{ here: } && \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: } && \text{ Bödigheimer 87: } \\ \\ Conf(\mathbb{R}^d,Y) &=& C_d( Y/\partial Y ) &=& C( \mathbb{R}^d, \emptyset; Y ) \\ \mathrm{Conf}_n\left( \mathbb{R}^d \right) & = & F_n C_d( S^0 ) / F_{n-1} C_d( S^0 ) & = & D_n\left( \mathbb{R}^d, \emptyset; S^0 \right) \\ \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right) & = & F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y ) & = & D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y \right) \\ \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0 \right) \\ \mathrm{Conf}_n( X, Y ) && &=& D_n\left( X, \partial X; Y/\partial Y \right) }

Notice here that when YY happens to have empty boundary, Y=\partial Y = \emptyset, then the pushout

Y/YYY* Y / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast

is YY with a disjoint basepoint attached. Notably for Y=*Y =\ast the point space, we have that

*/*=S 0 \ast/\partial \ast = S^0

is the 0-sphere.

Statements

Prelude: Equivalence to the infinite configuration space

First recall the following equivalence already before stabilization:

Proposition

For

  1. dd \in \mathbb{N}, d1d \geq 1 a natural number with d\mathbb{R}^d denoting the Cartesian space/Euclidean space of that dimension,

  2. YY a manifold, with non-empty boundary so that Y/YY / \partial Y is connected,

the scanning map constitutes a homotopy equivalence

Conf( d,Y)scanΩ dΣ d(Y/Y) Conf\left( \mathbb{R}^d, Y \right) \overset{scan}{\longrightarrow} \Omega^d \Sigma^d (Y/\partial Y)

between

  1. the configuration space of arbitrary points in d×Y\mathbb{R}^d \times Y vanishing at the boundary (Def. )

  2. the dd-fold loop space of the dd-fold reduced suspension of the quotient space Y/YY / \partial Y (regarded as a pointed topological space with basepoint [Y][\partial Y]).

In particular when Y=𝔻 kY = \mathbb{D}^k is the closed ball of dimension k1k \geq 1 this gives a homotopy equivalence

Conf( d,𝔻 k)scanΩ dS d+k Conf\left( \mathbb{R}^d, \mathbb{D}^k \right) \overset{scan}{\longrightarrow} \Omega^d S^{ d + k }

with the dd-fold loop space of the (d+k)-sphere.

(May 72, Theorem 2.7, Segal 73, Theorem 3)

Stable splitting of mapping spaces

Proposition

(stable splitting of mapping spaces out of Euclidean space/n-spheres)

For

  1. dd \in \mathbb{N}, d1d \geq 1 a natural number with d\mathbb{R}^d denoting the Cartesian space/Euclidean space of that dimension,

  2. YY a manifold, with non-empty boundary so that Y/YY / \partial Y is connected,

there is a stable weak homotopy equivalence

Σ Conf( d,Y)nΣ Conf n( d,Y) \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)

between

  1. the suspension spectrum of the configuration space of an arbitrary number of points in d×Y\mathbb{R}^d \times Y vanishing at the boundary and distinct already as points of d\mathbb{R}^d (Def. )

  2. the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in d×Y\mathbb{R}^d \times Y, vanishing at the boundary and distinct already as points in d\mathbb{R}^d (also Def. ).

Combined with the stabilization of the scanning map homotopy equivalence from Prop. this yields a stable weak homotopy equivalence

(4)Maps cp( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))=Ω dΣ d(Y/Y)Σ scanΣ Conf( d,Y)nΣ Conf n( d,Y) Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) = \Omega^d \Sigma^d (Y/\partial Y) \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow} \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)

between the latter direct sum and the suspension spectrum of the mapping space of pointed continuous functions from the d-sphere to the dd-fold reduced suspension of Y/YY / \partial Y.

(Snaith 74, theorem 1.1, Bödigheimer 87, Example 2)

In fact by Bödigheimer 87, Example 5 this equivalence still holds with YY treated on the same footing as d\mathbb{R}^d, hence with Conf n( d,Y)Conf_n(\mathbb{R}^d, Y) on the right replaced by Conf n( d×Y)Conf_n(\mathbb{R}^d \times Y) in the well-adjusted notation of Def. :

Maps cp( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))nΣ Conf n( d×Y) Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)

In terms of Goodwillie-Taylor towers

We discuss the interpretation of the above stable splitting of mapping spaces from the point of view of Goodwillie calculus, following Arone 99, p. 1-2, Goodwillie 03, p. 6.

Observe that the configuration space of points Conf n(X,Y)Conf_n(X,Y) from Def. , given by the formula (3)

Conf n(X,Y)(((X nΔ X n)×Y n)/(X n×Y n))/Σ(n) Conf_n(X,Y) \;\coloneqq\; \Big( \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^n \big) / \partial(X^n \times Y^n) \Big) /\Sigma(n)

is the quotient by the symmetric group-action of the smash product Conf n(X)(Y/Y) nConf_n(X) \wedge (Y/\partial Y)^n of the plain Configuration space Conf n(X)Conf_n(X) (2) (regarded as a pointed topological space with basepoint the class of the boundary [(X n)]\left[\partial\left(X^n\right)\right]) with the analogous pointed topological space given by YY, the latter in fact being (since here we do not form the complement by the fat diagonal) an nn-fold smash product itself:

Y × n/(Y × n)(Y/Y) n. Y^{\times_n}/\partial (Y^{\times_n}) \;\simeq\; ( Y/\partial Y )^{\wedge_n} \,.

Hence in summary:

(5)Conf n(X,Y)Conf n ord(X) Σ(n)(Y/Y) n, Conf_n(X, Y) \;\simeq\; Conf^{ord}_n(X) \wedge_{\Sigma(n)} \left( Y/\partial Y \right)^{\wedge_n} \,,

where

Conf n ord(X)(X × nΔ X n)/(X n) Conf_n^{ord}(X) \;\coloneqq\; \left( X^{\times_n} \setminus \mathbf{\Delta}_X^n \right)/ \partial(X^n)

is the ordered configuration space (1).

This construction, regarded as a functor from pointed topological spaces to spectra

Top */ Spectra Z Σ Conf n ord(X) Σ(n)Z n \array{ Top^{\ast/} &\longrightarrow& Spectra \\ Z &\mapsto& \Sigma^\infty Conf^{ord}_n(X) \wedge_{\Sigma(n)} Z^{\wedge_n} }

is an n-homogeneous (∞,1)-functor in the sense of Goodwillie calculus, and hence the partial wedge sums as nn ranges

(6)Zk{1,,n}Σ Conf k ord(X) Σ(k)Z k Z \;\mapsto\; \underset{k \in \{1, \cdot, n\}}{\bigoplus} \Sigma^\infty Conf^{ord}_k(X) \wedge_{\Sigma(k)} Z^{\wedge_k}

are n-excisive (∞,1)-functors. Moreover, by the stable splitting of mapping spaces (4) of Prop. , there is a projection morphism onto the first nn wedge summands

(7)Maps cp( d,Σ dZ) = Maps */(S d,Σ dZ) kΣ Conf k ord( d) Σ(k)Z k p n k{1,,n}Σ Conf k ord( d) Σ(k)Z k \array{ Maps_{cp}(\mathbb{R}^d, \Sigma^d Z) &=& Maps^{\ast/}( S^d, \Sigma^d Z) &\simeq& \underset{k \in \mathbb{N}}{\oplus} \Sigma^\infty Conf^{ord}_k(\mathbb{R}^d) \wedge_{\Sigma(k)} Z^{\wedge_k} \\ && && \Big\downarrow {}^{\mathrlap{ p_n }} \\ && && \underset{k \in \{1, \cdot, n\}}{\bigoplus} \Sigma^\infty Conf^{ord}_k( \mathbb{R}^d ) \wedge_{\Sigma(k)} Z^{\wedge_k} }

and this is (n+1)k-connected when ZZ is k-connected.

By Goodwillie calculus this means that (6) are, up to equivalence, the stages

(8)P nMaps */(S d,Σ d()):Zk{1,,n}Σ Conf k ord(S d,Z) P_n Maps^{\ast/}( S^d, \Sigma^d (-)) \;\colon\; Z \mapsto \underset{k \in \{1, \cdot, n\}}{\bigoplus} \Sigma^\infty Conf^{ord}_k(S^d, Z)

at ZTop */Z \in Top^{\ast/} of the Goodwillie-Taylor tower for the mapping space-functor

Maps cp( d,Σ d())=Maps */(S d,Σ d()):Top */Top */. Maps_{cp}(\mathbb{R}^d, \Sigma^d (-)) = Maps^{\ast/}( S^d, \Sigma^d (-)) \;\colon\; Top^{\ast/} \longrightarrow Top^{\ast/} \,.

Therefore the stable splitting theorem may equivalently be read as expressing the mapping space functor equivalently as the limit over its Goodwillie-Taylor tower.

(Arone 99, p. 1-2, Goodwillie 03, p. 6)

\,

Lax closed structure on Σ \Sigma^\infty

Notice that the first stage in the Goodwillie-Taylor tower of Maps(S d,Σ d())Maps(S^d, \Sigma^d(-)) is

P 1Maps */(S d,Σ d(Y/Y)) =Σ Conf 1 ord( d,Y) Σ Conf 1 ord( d)S 0(Y/Y) Σ (Y/Y) Ω dΣ dΣ (Y/Y) Maps(Σ S d,Σ d(Y/Y)) \begin{aligned} P_1 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y) ) & = \Sigma^\infty Conf^{ord}_1( \mathbb{R}^d , Y ) \\ & \simeq \Sigma^\infty \underset{\simeq S^0}{\underbrace{Conf^{ord}_1( \mathbb{R}^d )}} \wedge (Y/\partial Y) \\ & \simeq \Sigma^\infty (Y/\partial Y) \\ & \simeq \Omega^d \Sigma^d \Sigma^\infty (Y/\partial Y) \\ & \simeq Maps\left( \Sigma^\infty S^d, \Sigma^d (Y/\partial Y) \right) \end{aligned}

Here in the first step we used (8), in the second step we used (5). Under the brace we observe that space of configurations of a single point in d\mathbb{R}^d is trivially d\mathbb{R}^d itself, which is contractible d*\mathbb{R}^d \simeq \ast and, due to empty boundary of d\mathbb{R}^d, contributes a 0-sphere-factor to the smash product, which disappears. In the last last two steps we trivially rewrote the result to exhibit it as a mapping spectrum.

Therefore the projection p 1p_1 (7) to the first stage of the Goodwillie-Taylor tower is of the form

p 1:Σ Maps(S d,Σ d(Y/Y))Maps(Σ S d,Σ Σ d(Y/Y)). p_1 \;\colon\; \Sigma^\infty Maps\left( S^d , \Sigma^d (Y /\partial Y) \right) \longrightarrow Maps \left( \Sigma^\infty S^d, \Sigma^\infty \Sigma^d (Y / \partial Y) \right) \,.

Since Σ \Sigma^\infty is a strong monoidal functor (here), there is a canonical comparison morphism of this form, exhibiting the induce lax closed-structure on Σ \Sigma^\infty. Probably p 1p_1 coincides with that canonical morphism, up to equivalence.

Does it?

References

The theorem is originally due to

  • Victor Snaith, A stable decomposition of Ω nS nX\Omega^n S^n X, Journal of the London Mathematical Society 7 (1974), 577 - 583 (pdf)

using the homotopy equivalence before stabilization due to

  • Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf)

  • Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)

An alternative proof is due to

Review and generalization is due to

Interpretation in terms of the Goodwillie-Taylor tower of mapping spaces is due to

  • Greg Arone, A generalization of Snaith-type filtration, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (pdf)

  • Michael Ching, Calculus of Functors and Configuration Spaces, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (pdf)

  • Thomas Goodwillie, p. 6 of Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711 (journal, arXiv:math/0310481))

Generalization via nonabelian Poincaré duality from codomains which are nn-fold suspensions to general n-connective spaces:

  • Lauren Bandklayder, Stable splitting of mapping spaces via nonabelian Poincaré duality (arxiv:1705.03090)

Last revised on October 28, 2019 at 18:13:34. See the history of this page for a list of all contributions to it.