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Representing Homology Classes

Representing Homology Classes C. P. ROURKE This note is concerned with the problem of finding simple geometric representa- tives for homology classes. Thorn [6] has proved that any Z -homology class is represented by a manifold, while his counterexample can be used to show that Z -classes are not representable by Z -manifolds (in the sense of Sullivan) for p ^ 2. p p We will show that any Z -class is represented by a "p-polyhedron " which is a generalised twisted Z -manifold (see §2); this answers a conjecture of Sullivan. We conjecture that p-polyhedra are in fact the simplest general representatives for Z -homology. The main theorem, from which the representation result follows easily, is that any connected ring theory which represents the cohomology of the standard p-Lens space represents Z -(co)homology in general. The theorem is proved by a short spectral sequence argument using a result of Serre on the cohomology of K(l , n). Since the 2-Lens space is real projective space, we recover (a strengthened form of) Thorn's result (and the whole proof is considerably simpler and has less algebraic input than Thorn's original one). 1. Definitions and statement of the main theorem Let p be a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Representing Homology Classes

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/5.3.257
Publisher site
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Abstract

C. P. ROURKE This note is concerned with the problem of finding simple geometric representa- tives for homology classes. Thorn [6] has proved that any Z -homology class is represented by a manifold, while his counterexample can be used to show that Z -classes are not representable by Z -manifolds (in the sense of Sullivan) for p ^ 2. p p We will show that any Z -class is represented by a "p-polyhedron " which is a generalised twisted Z -manifold (see §2); this answers a conjecture of Sullivan. We conjecture that p-polyhedra are in fact the simplest general representatives for Z -homology. The main theorem, from which the representation result follows easily, is that any connected ring theory which represents the cohomology of the standard p-Lens space represents Z -(co)homology in general. The theorem is proved by a short spectral sequence argument using a result of Serre on the cohomology of K(l , n). Since the 2-Lens space is real projective space, we recover (a strengthened form of) Thorn's result (and the whole proof is considerably simpler and has less algebraic input than Thorn's original one). 1. Definitions and statement of the main theorem Let p be a

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1973

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