## Smooth structure and signature of codimension $2$ embeddings

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- by Dieter Erle
- Proc. Amer. Math. Soc.
**35**(1972), 611-616 - DOI: https://doi.org/10.1090/S0002-9939-1972-0303553-X
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## 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.

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## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**35**(1972), 611-616 - MSC: Primary 57D40
- DOI: https://doi.org/10.1090/S0002-9939-1972-0303553-X
- MathSciNet review: 0303553