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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Highly connected embeddings in codimension two

Author: Susan Szczepanski
Journal: Trans. Amer. Math. Soc. 280 (1983), 139-159
MSC: Primary 57R40; Secondary 57R67
MathSciNet review: 712253
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Abstract: In this paper we study semilocal knots over $ f$ into $ \xi $, that is, embeddings of a manifold $ N$ into $ E(\xi)$, the total space of a $ 2$-disk bundle over a manifold $ M$, such that the restriction of the bundle projection $ p:E(\xi) \to M$ to the submanifold $ N$ is homotopic to a normal map of degree one, $ f:N \to N$. We develop a new homology surgery theory which does not require homology equivalences on boundaries and, in terms of these obstruction groups, we obtain a classification (up to cobordism) of semilocal knots over $ f$ into $ \xi $. In the simply connected case, the following geometric consequence follows from our classification. Every semilocal knot of a simply connected manifold $ M\char93 K$ in a bundle over $ M$ is cobordant to the connected sum of the zero section of this bundle with a semilocal knot of the highly connected manifold $ K$ into the trivial bundle over a sphere.

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Keywords: Surgery, cobordism of embeddings, Hermitian forms, highly connected maps and manifolds
Article copyright: © Copyright 1983 American Mathematical Society