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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Groups of free involutions of homotopy $S^{[n/2]}\times S^{[(n+1)/2]}$’s
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by H. W. Schneider PDF
Trans. Amer. Math. Soc. 206 (1975), 99-136 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 206 (1975), 99-136
  • MSC: Primary 57E25; Secondary 57D60
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0370635-3
  • MathSciNet review: 0370635