nLab stable splitting of mapping spaces

Idea

The stabilization/suspension spectrum $\Sigma^\infty Maps(X,A)$ of mapping spaces $Maps(X,A)$ between suitable CW-complexes $X, A$ 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 $Maps(X,-)$ and often expressible in terms of the configuration spaces of $X$.

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 $X$ be a manifold, possibly with boundary.

For $n \in \mathbb{N}$, the configuration space of $n$ distinguishable points in $X$ disappearing at the boundary is the topological space

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

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

Then the configuration space of $n$ in-distinguishable points in $X$ is the further quotient topological space

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

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

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

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

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

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

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

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

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

by the equivalence relation $\sim$ given by

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

This is naturally a filtered topological space with filter stages

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

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

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

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:

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

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

$$Y / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast$$

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

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

is the 0-sphere.

Statements

Prelude: Equivalence to the infinite configuration space

First recall the following equivalence already before stabilization:

Proposition

For

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

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

the scanning map constitutes a homotopy equivalence

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

between

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

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

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

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

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

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

there is a stable weak homotopy equivalence

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

between

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

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

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

(4)$$\Sigma^\infty \, \Omega^d \Sigma^d (Y/\partial Y) \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow} \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)$$

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

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

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

$$\Sigma^\infty \, Maps_{cp}\big(\mathbb{R}^d, \Sigma^d (Y / \partial Y)\big) = \Sigma^\infty \, Maps^{\ast/}\big( S^d, \Sigma^d (Y / \partial Y)\big) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y) \,.$$

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 $Conf_n(X,Y)$ from Def. , given by the formula (3)

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

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

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

Hence in summary:

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

where

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

is the ordered configuration space (1).

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

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

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

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

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

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

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

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

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

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

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

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

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

$\,$

Lax closed structure on $\Sigma^\infty$

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

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

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 $\mathbb{R}^d$ is trivially $\mathbb{R}^d$ itself, which is contractible $\mathbb{R}^d \simeq \ast$ and, due to empty boundary of $\mathbb{R}^d$, 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 $p_1$ (7) to the first stage of the Goodwillie-Taylor tower is of the form

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

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

Does it?

Related concepts

• mapping spectrum

References

The theorem is originally due to

• Victor Snaith, A stable decomposition of $\Omega^n S^n X$, 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

• Ralph Cohen, Stable proof of stable splittings, Math. Proc. Camb. Phil. Soc., 1980, 88, 149 (doi:10.1017/S030500410005742X, pdf)

Review and generalization:

• Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, in: Algebraic topology, Lecture Notes in Mathematics 1286, Springer (1987) 174-187 [pdf, pdf, doi:10.1007/BFb0078741]

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))

A proof via nonabelian Poincaré duality:

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

See also:

• Douglas Ravenel, What we still don’t understand about loop spaces of spheres, Contemporary Mathematics 1998 (pdf, pdf)