Transactions of the American Mathematical Society

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



Adjacent connected sums and torus actions

Author: Dennis McGavran
Journal: Trans. Amer. Math. Soc. 251 (1979), 235-254
MSC: Primary 57S25; Secondary 57N15, 57Q15, 57R05
MathSciNet review: 531977
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Abstract: Let M and N be closed, compact manifolds of dimension m and let X be a closed manifold of dimension $ n < m$ with embeddings of $ X\, \times \,{D^{m - n}}$ into M and N. Suppose the interior of $ X\, \times \,{D^{m - n}}$ is removed from M and N and the resulting manifolds are attached via a homeomorphism $ f:\,X \times \,{S^{m - n - 1}}\, \to \,X\, \times \,{S^{m - n - 1}}$. Let this homeomorphism be of the form $ f(x,\,t)\, = \,(x,\,F(x)(t))$ where $ F:\,X \to \,SO(m - n)$. The resulting manifold, written as $ M\,{\char93 _X}\,N$, is called the adjacent connected sum of M and N along X. In this paper definitions and examples are given and the examples are then used to classify actions of the torus $ {T^n}$ on closed, compact, connected, simply connected $ (n\, + \,2)$-manifolds, $ n \geqslant \,4$.

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Keywords: Adjacent connected sums, torus actions, simply connected manifolds, orbit space
Article copyright: © Copyright 1979 American Mathematical Society