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Über den Homotopietyp von Linsenraumprodukten. (German) Zbl 0588.55004
The (2n-1)-dimensional lens space $$L(m;r_ 1,...,r_ n)$$ is the quotient of $$S^{2n-1}\subset {\mathbb{C}}^ n$$ by a coordinatewise action of the cyclic group of order m, where a generator acts on the j’th coordinate as multiplication by $$\exp (2\pi ir_ j/m)$$. This paper addresses the question of when two finite products of (2n-1)-dimensional lens spaces $$(n>1)$$ are homotopy equivalent. An obvious necessary condition is, of course, that the fundamental groups (which are finite abelian groups) be isomorphic. Here a second necessary condition is given in terms of a congruence relation modulo the highest common factor of the various values of m of the terms in the product (when n is odd), or a collection of congruence relations modulo the prime-power divisors of this number (when n is even). The sufficiency or otherwise of these conditions is not discussed, but reference is made to a future paper of the first author for sufficiency in the case $$n=2$$.
Reviewer: J.Howie

##### MSC:
 55P15 Classification of homotopy type 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. 57R19 Algebraic topology on manifolds and differential topology
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