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Transactions of the American Mathematical Society

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

 
 

 

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


Author: H. W. Schneider
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
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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