Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
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 $X$.
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.
(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
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
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
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:
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
by the equivalence relation $\sim$ given by
This is naturally a filtered topological space with filter stages
The corresponding quotient topological spaces of the filter stages reproduces the above configuration spaces of a fixed number of points:
(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:
Notice here that when $Y$ happens to have empty boundary, $\partial Y = \emptyset$, then the pushout
is $Y$ with a disjoint basepoint attached. Notably for $Y =\ast$ the point space, we have that
is the 0-sphere.
First recall the following equivalence already before stabilization:
For
$d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,
$Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,
the scanning map constitutes a homotopy equivalence
between
the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. )
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
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 out of Euclidean space/n-spheres)
For
$d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,
$Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,
there is a stable weak homotopy equivalence
between
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. )
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
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. :
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)
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:
Hence in summary:
where
is the ordered configuration space (1).
This construction, regarded as a functor from pointed topological spaces to spectra
is an n-homogeneous (∞,1)-functor in the sense of Goodwillie calculus, and hence the partial wedge sums as $n$ ranges
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
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
at $Z \in Top^{\ast/}$ of the Goodwillie-Taylor tower for the mapping space-functor
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)
$\,$
Notice that the first stage in the Goodwillie-Taylor tower of $Maps(S^d, \Sigma^d(-))$ is
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
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?
The theorem is originally due to
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))
A proof via nonabelian Poincaré duality:
See also: