Contents - 2017

• Cubical simplicial volume
  Clara Löh and Christian Plankl
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• Brouwer degree, domination of manifolds, and groups presentable by products
  Pierre de la Harpe
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Contents - 2014

• Poincaré's conjecture
  Klaus Volkert
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Contents - 2013

• Fake lens spaces
  Tibor Macko
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• Lens spaces in dimension three: a history
  Klaus Volkert
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• Poincaré's cube manifolds
  Klaus Volkert
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• Poincaré's homology sphere
  Klaus Volkert
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• Projective plane: a history
  Klaus Volkert
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• Space forms: a history
  Klaus Volkert
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• 1-manifolds
  Oleg Viro
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Contents - 2012

• Aspherical manifolds
  Wolfgang Lück
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Contents - 2011

• Super manifolds: an incomplete survey
  Henning Hohnhold, Stephan Stolz and Peter Teichner
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• Simplicial volume
  Clara Löh
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• Hirzebruch surfaces
  Diarmuid Crowley and Matthias Kreck
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• Bordism
  Taras Panov
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• Complex bordism
  Taras Panov
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• Formal group laws and genera
  Taras Panov
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• 5-manifolds: 1-connected
  Diarmuid Crowley
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Fake lens spaces

Tibor Macko

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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