A Weiss-Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots

TL;DR

The paper extends the Weiss–Williams pseudoisotopy framework to spaces of embeddings, establishing a Weiss–Williams–type map in the concordance embedding stable range and identifying its target with the infinite loop space $oldsymbol{CE}(P,M)_{hC_2}$. It leverages orthogonal calculus and trace methods from algebraic K-theory to relate the difference between block and ordinary embeddings to relative $A$-theory via $oldsymbol{CE}(P,M)$, yielding explicit connectivity and split fibre sequences in high codimensions. A detailed analysis of involutions in the $C_2$-equivariant $K$-theory landscape is developed, connecting $oldsymbol{H}(M)$ and $oldsymbol{Wh}^{ ext{Diff}}(M)$ through deformations and suspensions, and enabling transfer of information to the $A$-theory setting. Applying these tools, the authors provide a complete description of the homotopy type of spaces of long knots $ ext{Emb}_ullet(D^p,D^d)$ for $d-p\nge 3$, localized away from 2, in the concordance embedding stable range, and compute new torsion information in the relevant homotopy groups. The results illuminate the structure of the Gromoll filtration and connect high-dimensional knot spaces to stable $h$-cobordism theory, with potential implications for the hairy graph complex in rational homotopy theory and for explicit computations of low-dimensional homotopy groups of long knots.

Abstract

We establish a pseudoisotopy result for embedding spaces in the line of that of Weiss and Williams for diffeomorphism groups. In other words, for $P\subset M$ a codimension at least three embedding, we describe the difference in a range of homotopical degrees between the spaces of block and ordinary embeddings of $P$ into $M$ as a certain infinite loop space involving the relative algebraic $K$-theory of the pair $(M,M-P)$. This range of degrees is the so-called concordance embedding stable range, which, by recent developments of Goodwillie-Krannich-Kupers, is far beyond that of the aforementioned theorem of Weiss-Williams. We use this result to obtain split fibre sequences in the concordance embedding stable range, with explicit, analysable base and fibre, which determine the homotopy type of spaces of long knots of codimension at least 3. This leads to explicit computations of homotopy groups, including torsion information, in that range. In doing so, we carry out an extensive analysis of certain geometric involutions in algebraic $K$-theory that may be of independent interest.

Figures (11)

• Figure 1: Images of $\gamma\in \Omega C(M)$ under the graphing maps $\Gamma$ and $U\Gamma$, and the homotopy between them. The concordances are equal to the identity on grey shaded regions.
• Figure 2: Depiction of the topological manifold $\widehat{a}(\rho)$ for $\rho\in H^s(M)$.
• Figure 3: Paths $\gamma_{W}^*$ and $\gamma_{V^*}$ in $B\mathrm{Diff}^{b}_{\partial}(M\times \underline{\mathbb{R}})$. The arrow indicates the direction of the path as time increases.
• Figure 4: Depiction of the manifold $U_\rho$. Proceed with caution: the part of the picture corresponding to $N$ takes place in an extra dimension that we are unable to depict accurately.
• Figure 5: $E_2$-page of the Adams spectral sequence at the prime $3$ for $\mathbb{S}^{\rho_3}_{hD_3}$ when $d-p=3$. Here $s$ denotes the degree in the Adams filtration and $t-s$ is the total degree.

Theorems & Definitions (118)

• Theorem 1.1: Weiss--Williams
• Theorem A
• Remark 1.2
• Remark 1.3: Splitting results and the Gromoll filtration
• Remark 1.4: Topological version of Theorem \ref{['EmbWWIThm']}
• Remark 1.5
• Theorem B
• Remark 1.6
• Corollary C: Propositions \ref{['htpygroupsEmcoprimeprop']} & \ref{['htpyat3prop']}
• Remark 2.1