SPIN STRUCTURES AND CODIMENSION-TWO HOMEOMORPHISM EXTENSIONS

Let i : M hooked right arrow Rp+2 be a smooth embedding from a connected, oriented, closed p-dimesional smooth manifold to Rp+2, then there is a spin structure i#(zeta(p+2)) on M canonically induced from the embedding. If an orientation-preserving diffeomorphism tau of M extends over i as an orientation-preserving topological homeomorphism of Rp+2, then tau preserves the induced spin structure. For C being Top, PL or Diff, let E-C(i) be the subgroup of the C-mapping class group MCG(C)(M) consisting of elements whose representatives extend over Rp+2 as orientation- preserving C-homeomorphisms. We apply the invariance of i(#)(zeta(p+2)) to study left perpendicularMCG(C)(M) : E-C(i)right perpendicular when M is a p-dimensional torus or a closed-orientable surface.