Mathematical Research Letters

Volume 19 (2012)

Number 2

Spin structures and codimension-two homeomorphism extensions

Pages: 345 – 357

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n2.a7

Authors

Fan Ding (School of Mathematical Sciences, Peking University, Beijing 100871, China.)

Yi Liu (University of California at Berkeley, CA 94720-3840, U.S.A.)

Shicheng Wang (School of Mathematical Sciences, Peking University, Beijing 100871, China.)

Jiangang Yao (Department of Mathematics, 970 Evans Hall, University of California at Berkeley, CA 94720-3840, U.S.A.)

Abstract

Let $\imath: M\hookrightarrow \RR^{p+2}$ be a smooth embeddingfrom a connected, oriented, closed $p$-dimesionalsmooth manifold to $\RR^{p+2}$,then there is a spin structure $\imath^\sharp(\varsigma^{p+2})$ on $M$canonically induced from the embedding. If anorientation-preserving diffeomorphism $\tau$ of $M$ extends over$\imath$ as an orientation-preserving topological homeomorphism of$\RR^{p+2}$, then $\tau$ preserves the induced spin structure.

For $\cat$ being $\topo$, $\pl$ or $\diff$,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.We apply the invariance of $\imath^\sharp(\varsigma^{p+2})$ to study$[\mcg_\cat(M):\esg_\cat(\imath)]$ when $M$ isa $p$-dimensional torus or a closed-orientable surface.

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