Goodwillie–Weiss Embedding Calculus

• Goodwillie–Weiss embedding calculus is a homotopical framework that approximates smooth and Poincaré embeddings via a Taylor tower of polynomial approximations.
• It leverages configuration spaces, operadic structures, and duality theories to bridge classical obstruction theory with manifold topology.
• The framework provides practical insights into interpolating between stable normal invariants, immersions, and unlinked embeddings with strong convergence guarantees.

The Goodwillie–Weiss embedding calculus is a homotopical framework for systematically approximating spaces of smooth (or block, or topological) embeddings between manifolds via a sequence of functorial polynomial approximations, termed the Taylor tower. The theory provides both convergent and non-convergent polynomial towers whose layers are determined by configuration spaces, operadic structures, and, in significant extensions, Poincaré duality spaces, yielding connections to classical obstruction theory, manifold topology, and functor calculus. The embedding calculus admits variants adapted to Poincaré duality spaces, providing robust tools for analyzing the space of codimension-zero embeddings and their interpolation with stable phenomena such as normal invariants and immersions (Klein, 2014).

1. Poincaré Embedding Spaces and Unstable Normal Invariants

Let $P$ be an orientable $n$-dimensional Poincaré space with boundary %%%%2%%%%, $\partial P \to P$ a cofibration and at least $2$-connected (homotopy-codimension $\geq 3$). The space $E(P,D^n)$ parametrizes codimension-zero Poincaré embeddings of $P$ into $D^n$, formalized via data of an $n$-dimensional Poincaré complement $C$ and a boundary-gluing map $\partial P \sqcup S^{n-1} \to C$ such that the homotopy pushout $P \leftarrow \partial P \to C$ is weakly contractible.

A sectioned Poincaré space is a cofibrant model $K \simeq P$ with section $s: K \to \partial P$, leading to the generalized Thom space $P^{\sectionmark} := \partial P \cup_{K} CK$, weakly equivalent to the (de)suspension of $P/\partial P$. An unstable normal invariant for $P$ is a based map $\alpha: S^{n-1} \to P^{\sectionmark}$ sending the fundamental class $[S^{n-1}]$ to a fundamental class in $H_n(P,\partial P)$. The space of such invariants is $\Omega^{n-1}P^{\sectionmark}$.

The Browder construction assigns to an unstable normal invariant $\alpha$ a Poincaré embedding in $D^n$, giving the Browder map

$\Omega^{n-1}P^{\sectionmark} \longrightarrow E(P,D^n).$

2. Unlinked Embeddings, Poincaré Immersions, and the Browder Equivalence

A Poincaré embedding $C$ is unlinked if the boundary-gluing map factors through $P^{\sectionmark}$, or equivalently, $C$ is equipped with a null-homotopy of its link map $P \to C$. The subspace of unlinked embeddings is $SE(P,D^n) \subset E(P,D^n)$, and the refined Browder map is a homotopy equivalence: $\Omega^{n-1}P^{\sectionmark} \simeq SE(P,D^n).$

Poincaré immersion spaces are obtained by iterated decompression: $$I(P,D^n) := \operatorname{hocolim}\big[ E(P,D^n) \to E(P \times D^1, D^{n+1}) \to \cdots \big]$$ with a natural weak equivalence $I(P,D^n) \simeq \Omega^n Q(P/\partial P)$, the space of stable normal invariants, or equivalently, $I(P,D^n) \simeq F(P, G)$ where $G$ is the stable sphere self-equivalence space.

3. The Interpolating Tower: From Immersions to Unlinked Embeddings

Given a handle-dimension bound $k$ with $P$ admitting a handle decomposition up to index $k$, the homotopy-codimension condition $n-k\geq 3$ enables the construction of a tower of fibrations interpolating between Poincaré immersions and unlinked embeddings: $\cdots \to CE_2(P,D^n) \to CE_1(P,D^n)$ with compatible maps $\varphi_j: SE(P,D^n) \to CE_j(P,D^n)$. The initial stage $CE_1(P,D^n) \simeq I(P,D^n)$, and the inverse limit recovers the unlinked embedding space: $\varprojlim_j\, CE_j(P,D^n) \simeq SE(P,D^n).$

Each stage is defined by a homotopy pullback involving the Goodwillie functor calculus tower for the identity functor $I: \mathrm{Top}_* \to \mathrm{Top}_*$. For the generalized Thom space $P^{\sectionmark}$, the Goodwillie tower yields functors $P_j I(P^{\sectionmark})$ with layers: $L_j I(P^{\sectionmark}) \simeq \Omega^\infty \left( W_j \wedge (P^{\sectionmark})^{\wedge j} \right)_{h\Sigma_j}$ where $W_j$ is the $j$th derivative spectrum of the identity.

The $j$th stage $CE_j(P,D^n)$ is given as the homotopy pullback: $\begin{array}{ccc} CE_j(P,D^n) & \to & \Omega^n P_j I(P^{\sectionmark}) \ \downarrow & & \downarrow \ CE_{j-1}(P,D^n) & \to & \Omega^n P_{j-1} I(P^{\sectionmark}) \end{array}$ inducing the map $\varphi_j$ from $SE(P,D^n)$ to $CE_j(P,D^n)$.

4. Analysis of Layers, Connectivity, and Convergence

If $x \in CE_{j-1}(P,D^n)$ lifts to $CE_j$, then the space of lifts is homotopy equivalent to the infinite-loop space of

$\left( (P^{\sectionmark})^{\wedge j} \wedge W_j \right)_{h\Sigma_j}$

or, via Spanier–Whitehead duality, is described as

$$F_*(P^{\times j}_+,\,W_j \wedge S^{(n-1)V_j})_{h\Sigma_j}$$

with $V_j$ the reduced standard $\Sigma_j$-representation.

The connectivity of $\varphi_j$ satisfies

$$\operatorname{conn}(\varphi_j) \geq 2 - n + (j+1)(n - k - 2),$$

which increases with $j$ and ensures that the tower strongly converges to $SE(P,D^n)$ when $n - k - 2 > 0$.

5. Main Theorems and Their Proofs

Theorem A (Fiber of the Browder Map): For $C \in E(P,D^n)$, the homotopy fiber of $\Omega^{n-1}P^{\sectionmark} \to E(P,D^n)$ over $C$ is nonempty if and only if the link class $\ell o(C) \in [P^+, C]$ vanishes. If $\ell o(C)=0$, then

$F_*(P^+, C) \to \Omega^{n-1} P^{\sectionmark} \to E(P,D^n)$

is a fiber sequence at the chosen null-homotopy.

Theorem B (Refined Browder Equivalence): The refined Browder map $\Omega^{n-1} P^{\sectionmark} \to SE(P,D^n)$ is a homotopy equivalence. This identifies unlinked embeddings as precisely the data of an unstable normal invariant factoring through $P^{\sectionmark}$.

Theorem C (The Poincaré Embedding Tower): Under the homotopy-codimension condition, the tower

$\cdots \to CE_2(P,D^n) \to CE_1(P,D^n)$

has

• $$\operatorname{conn}(\varphi_j) \geq 2-n + (j+1)(n-k-2)$$,
• $CE_1(P,D^n) \simeq I(P,D^n)$,
• $\varprojlim_j\,CE_j(P,D^n)\simeq SE(P,D^n)$,
• layers $\operatorname{fib}(CE_j\to CE_{j-1})\simeq\Omega^\infty(W_j\wedge(P^{\sectionmark})^{\wedge j})_{h\Sigma_j}$.

Proofs combine properties of Goodwillie’s tower, model-category calculations for the relevant section spaces, and duality theorems to translate between these models.

6. Conjectural Comparison with Manifold Calculus

For a compact smooth $(n-1)$-manifold $Q$ with handle-dimension $k\leq n-4$, there is a decompression/forgetful map from the smooth embedding space

$$\operatorname{Emb}^{sm}(Q,D^{n-1}) \to SE(Q\times D^1, D^n)$$

which is conjectured to lift to a map of towers from the manifold (Weiss) calculus tower for $\operatorname{Emb}^{sm}(Q,D^{n-1})$ to the Poincaré tower $CE_j(Q\times D^1, D^n)$.

At the $j$th layer, the smooth side consists of spaces of compactly supported sections over configuration spaces, $$\Gamma_c(\operatorname{Conf}(Q,j)/\Sigma_j;$$, and the Poincaré side is $\Omega^\infty((P^{\sectionmark})^{\wedge j}\wedge W_j)_{h\Sigma_j}$. There is a diagram linking these through Pontryagin–Thom collapse, stabilization, Adams equivalence, excision, and forget-the-diagonal arguments. The composite map increases in connectivity as $j\to\infty$, suggesting the Poincaré tower recovers the classical manifold-calculus tower in the smoothable case.

7. Relationship with Functor Calculus and Operadic Derivatives

The Poincaré version of embedding calculus is governed by the Goodwillie calculus of the identity functor $I: \mathrm{Top}_* \to \mathrm{Top}_*$, with each layer in the interpolating tower expressed via the coefficient spectra (the derivatives $W_j$) of $I$. These spectra, equipped with inherited $\Sigma_j$-actions, reflect the universal target for polynomial approximations. The composition with the generalized Thom space yields explicit mapping space models for each layer.

The approach generalizes classical manifold calculus, leveraging model-category presentations, the Browder–Pontryagin–Thom construction, and Spanier–Whitehead duality to render the space of codimension-zero embeddings tractable even in Poincaré duality settings.

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The Goodwillie–Weiss embedding calculus for Poincaré duality spaces thus establishes a stable polynomial approximation tower interpolating between stable normal invariants (Poincaré immersions) and unlinked embeddings, with explicit layers governed by Goodwillie derivatives and connectivity estimates yielding strong convergence under codimension hypotheses. The theory provides a blueprint for connections to (and conjectural unification with) the usual manifold-calculus tower and operates as a bridge between high-dimensional embedding theory, stable homotopy theory, and functor calculus (Klein, 2014).