Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Smooth structure and signature of codimension $ 2$ embeddings


Author: Dieter Erle
Journal: Proc. Amer. Math. Soc. 35 (1972), 611-616
MSC: Primary 57D40
MathSciNet review: 0303553
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In an earlier paper (Topology 8 (1969), 99-114), the author defined a signature for codimension 2 embeddings in $ {S^{4k + 1}}$ and proved this signature to be an invariant of the topological type of the embedding. For embedded homotopy $ (4k - 1)$-spheres, this signature is known to detect the smooth structure. It turns out that it determines the smooth structure also for integer homology $ (4k - 1)$-spheres and, by a result of A. Durfee, for closed $ (2k - 2)$ connected $ (4k - 1)$-manifolds with finite $ (2k - 1)$-dimensional homology. As a consequence, in the above cases, the smooth structure is given by the topological type of the embedding. On the other hand, for $ k = 3,4,5,7,15$, we exhibit examples of $ (4k - 1)$ manifolds embedded in a $ (4k + 1)$-sphere for which the smooth structure is not determined by the signature of the embedding.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57D40

Retrieve articles in all journals with MSC: 57D40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0303553-X
Keywords: Smooth structure, signature, manifold, higher dimensional knots, codimension 2, topological type, embedding, homology sphere, highly connected manifold
Article copyright: © Copyright 1972 American Mathematical Society