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Fake lens spaces

Tibor Macko

Abstract

A fake lens space is an orbit space of a free action of a finite cyclic group on a sphere and as such it is a generalization of a classical lens space. The invariants of fake lens spaces described here are their homotopy groups, homology groups, a certain k-invariant, the Reidemeister torsion, the rho-invariant and certain splitting invariants. We survey the classification of fake lens spaces which includes the classification up to homotopy, up to simple homotopy and up to homeomorphism, employing methods of homotopy theory, algebraic K-theory and surgery theory. Finally we discuss the join construction which builds fake lens spaces from other fake lens spaces of a lower dimension.

Keywords

lens space, fake lens space, Reidemeister torsion, rho-invariant, splitting invariants, surgery, structure set, join construction

Mathematical Subject Classification 2010

Primary: 57R65, 57S25

Publication

Received: 31st March 2012
Revised: 4th January 2013, 8th March 2013
Accepted: 20th March 2013
Published: 25th April 2013
Editor: Ian Hambleton