Embeddings of the base and bundle isomorphisms
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- 作者：De Sapio ; Rodolfo
- 关键词：57R40 ; 57R57 ; Smooth embedding ; Oriented vector bundle ; Isomorphism
- 刊名：Topology and its Applications
- 出版年：2003
- 出版时间：May 15, 2003
- 年：2003
- 卷：130
- 期：3
- 页码：221-237
- 全文大小：183 K
文摘
Let πi:Ei→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections si:M→Ei. Suppose that U is an open neighborhood of s1(M) in E1 and F:U→E2 a smooth embedding so that π2
Fs1:M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E1 and the induced bundle f*E2 are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<k≤n we give examples of nonisomorphic oriented bundles E1 and E2 of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E1 and f*E2 are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.
Fs1:M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E1 and the induced bundle f*E2 are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<k≤n we give examples of nonisomorphic oriented bundles E1 and E2 of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E1 and f*E2 are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.