[Paper Review] Spin structures and codimension-two homeomorphism extensions

Fan Ding, Yi Liu+2 more|arXiv (Cornell University)|Oct 26, 2009

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper establishes a topological invariant—spin structures induced by codimension-two embeddings in Euclidean space—that constrains which mapping classes of manifolds extend over the ambient space. By showing that orientation-preserving diffeomorphisms extending over ℝ^{p+2} preserve the induced spin structure, the authors derive nontrivial lower bounds on the index of the extendable mapping class subgroup. The key result is that for unknotted surfaces in ℝ⁴ and certain tori in ℝ^{p+2}, these bounds are sharp, with indices exactly 2^{2g−1} + 2^{g−1} and 2^p − 1, respectively.

ABSTRACT

Let $\imath: M o \RR^{p+2}$ be a smooth embedding from a connected, oriented, closed $p$-dimesional smooth manifold to $\RR^{p+2}$, then there is a spin structure $\imath^\sharp(ς^{p+2})$ on $M$ canonically induced from the embedding. If an orientation-preserving diffeomorphism $τ$ of $M$ extends over $\imath$ as an orientation-preserving topological homeomorphism of $\RR^{p+2}$, then $τ$ preserves the induced spin structure. Let $\esg_\cat(\imath)$ be the subgroup of the $\cat$-mapping class group $\mcg_\cat(M)$ consisting of elements whose representatives extend over $\RR^{p+2}$ as orientation-preserving $\cat$-homeomorphisms, where $\cat= opo$, $\pl$ or $\diff$. The invariance of $\imath^\sharp(ς^{p+2})$ gives nontrivial lower bounds to $[\mcg_\cat(M):\esg_\cat(\imath)]$ in various special cases. We apply this to embedded surfaces in $\RR^4$ and embedded $p$-dimensional tori in $\RR^{p+2}$. In particular, in these cases the index lower bounds for $\esg_ opo(\imath)$ are achieved for unknotted embeddings.

Motivation & Objective

• To understand which mapping classes of a closed, oriented, smooth p-manifold extend over ℝ^{p+2} as ambient homeomorphisms.
• To identify topological invariants that obstruct such extensions.
• To compute or bound the index of the subgroup of extendable mapping classes in the full mapping class group.
• To show that for unknotted embeddings, the derived lower bounds on the index are sharp.

Proposed method

• Define a canonical spin structure on a p-manifold M via pullback from the standard spin structure on ℝ^{p+2} under a smooth embedding 𝝳: M → ℝ^{p+2}.
• Prove that any orientation-preserving diffeomorphism of M extending over ℝ^{p+2} as a topological homeomorphism preserves the induced spin structure on M.
• Use the fact that the induced spin structure on M is the boundary of a spin structure on a Seifert hypersurface in ℝ^{p+2} to analyze its topological invariance.
• Apply the spin structure invariance to derive lower bounds on the index [MCG_C(M) : 𝔈_C(𝝳)] for C = Top, PL, Diff.
• Use known results on mapping class groups of surfaces and tori, particularly the structure of the Torelli group and the moduli group, to compute or bound the index.
• Show that for unknotted embeddings, the lower bounds are achieved, using constructions from Hirose and results on spin cobordism and Rokhlin's theorem.

Experimental results

Research questions

• RQ1 Which mapping classes of a closed, oriented, smooth p-manifold extend over ℝ^{p+2} as orientation-preserving homeomorphisms of the ambient space?
• RQ2 How can the induced spin structure on a manifold via a codimension-two embedding be used to obstruct extensions of diffeomorphisms?
• RQ3 What is the exact index of the extendable mapping class subgroup for unknotted embeddings of surfaces in ℝ⁴ and tori in ℝ^{p+2}?
• RQ4 Can the lower bounds derived from spin structure invariance be realized as exact values for specific embeddings?
• RQ5 How does the choice of embedding (e.g., unknotted vs. knotted) affect the extendability of mapping classes?

Key findings

• For any smooth embedding of a genus-g surface F_g into ℝ⁴, the index [MCG_Top(F_g) : 𝔈_Top(𝝳)] is at least 2^{2g−1} + 2^{g−1}.
• This lower bound is sharp for unknotted embeddings of F_g in ℝ⁴, meaning the index is exactly 2^{2g−1} + 2^{g−1} in these cases.
• For any smooth embedding of the p-torus T^p into ℝ^{p+2} with non-Lie-group induced spin structure, the index [MCG_Top(T^p) : 𝔈_Top(𝝳)] is at least 2^p − 1.
• This bound is achieved for unknotted embeddings of T^p in ℝ^{p+2}, so the index is exactly 2^p − 1.
• The induced spin structure on T^p from an unknotted embedding is never the Lie-group spin structure, which is crucial for the lower bound.
• For p ≤ 3, the index [MCG_C(T^p) : 𝔈_C(𝝳)] is exactly 2^p − 1 for C = Diff, PL, or Top, and is finite for all p.