Spin structures and codimension-two homeomorphism extensions

Abstract

Let $\imath: M\to \RR^{p+2}$ be a smooth embedding from a connected, oriented, closed $p$-dimesional smooth manifold to $\RR^{p+2}$, then there is a spin structure ${ı}^{♯}({\varsigma }^{p+2})$ on $M$ canonically induced from the embedding. If an orientation-preserving diffeomorphism $\tau$ of $M$ extends over $ı$ as an orientation-preserving topological homeomorphism of $\RR^{p+2}$, then $\tau$ 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 $\cat=\topo$, $\pl$ or $\diff$. The invariance of ${ı}^{♯}({\varsigma }^{p+2})$ 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 $\esg_\topo(\imath)$ are achieved for unknotted embeddings.

14 pages, 1 figure

References in corpus (3)

• On diffeomorphisms over surfaces trivially embedded in the 4-sphere
• Regular homotopy classes of immersions of 3-manifolds into 5-space
• Extending $T^p$ automorphisms over $\RR^{p+2}$ and realizing DE attractors

Cited by in corpus (3)

• Extending finite group actions on surfaces over $S^3$
• Extending $T^p$ automorphisms over $\RR^{p+2}$ and realizing DE attractors
• Maximum Orders of Cyclic and Abelian Extendable Actions on Surfaces