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Transactions of the American Mathematical Society

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



Groups of free involutions of homotopy $ S\sp{[n/2]}\times S\sp{[(n+1)/2]}$'s

Author: H. W. Schneider
Journal: Trans. Amer. Math. Soc. 206 (1975), 99-136
MSC: Primary 57E25; Secondary 57D60
MathSciNet review: 0370635
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Abstract: Let $ M$ be an oriented $ n$-dimensional manifold which is homotopy equivalent to $ {S^l} \times {S^{n - l}}$, where $ l$ is the greatest integer in $ n/2$. Let $ Q$ be the quotient manifold of $ M$ by a fixed point free involution. Associated to each such $ Q$ are a unique integer $ k\bmod {2^{\varphi (l)}}$, called the type of $ Q$, and a cohomology class $ \omega $ in $ {H^1}(Q;{Z_2})$ which is the image of the generator of the first cohomology group of the classifying space for the double cover of $ Q$ by $ M$. Let $ {I_n}(k)$ be the set of equivalence classes of such manifolds $ Q$ of type $ k$ for which $ {\omega ^{l + 1}} = 0$, where two such manifolds are equivalent if there is a diffeomorphism, orientation preserving if $ k$ is even, between them. It is shown in this paper that if $ n \geq 6$, then $ {I_n}(k)$ can be given the structure of an abelian group. The groups $ {I_8}(k)$ are partially calculated for $ k$ even.

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Keywords: Fixed point free involution, stably parallelizable manifold, double cover, surgery, projective space, Moore-Postnikov decomposition, Whitney procedure, twisted Euler class
Article copyright: © Copyright 1975 American Mathematical Society