Über den Homotopietyp von Linsenraumprodukten. (German) Zbl 0588.55004

Huck, Günther; Metzler, Wolfgang

Fundam. Math. 125, 11-22 (1985).

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

Keywords:

(2n-1)-dimensional lens space; finite products of (2n-1)-dimensional lens spaces; homotopy equivalent