Cohomology of preimages with local coefficients

Gonçalves, Daciberg Lima; Wong, Peter

doi:10.2140/agt.2006.6.1471

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Cohomology of preimages with local coefficients

Daciberg Lima Gonçalves and Peter Wong

Algebraic & Geometric Topology 6 (2006) 1471–1489

DOI: 10.2140/agt.2006.6.1471

arXiv: 0904.4177

Abstract

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mo class="MathClass-punc">,</mo><mi>N</mi></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi> <mo class="MathClass-rel">⊂</mo> <mi>N</mi></math>$ be compact smooth manifolds of dimensions $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi> <mo class="MathClass-bin">+</mo> <mi>k</mi><mo class="MathClass-punc">,</mo><mi>n</mi></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℓ</mi></math>$ , respectively. Given a map $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi> <mo class="MathClass-punc">:</mo> <mi>M</mi> <mo class="MathClass-rel">→</mo> <mi>N</mi></math>$ , we give homological conditions under which $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>g</mi></mrow><mrow><mo class="MathClass-bin">−</mo><mn>1</mn></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>B</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ has nontrivial cohomology (with local coefficients) for any map $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi></math>$ homotopic to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi></math>$ . We also show that a certain cohomology class in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>H</mi></mrow><mrow><mi>j</mi></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>N</mi><mo class="MathClass-punc">,</mo><mi>N</mi><mo class="MathClass-bin">−</mo><mi>B</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is Poincaré dual (with local coefficients) under $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>f</mi></mrow><mrow><mo class="MathClass-bin">∗</mo></mrow></msup></math>$ to the image of a corresponding class in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo class="MathClass-bin">+</mo><mi>k</mi><mo class="MathClass-bin">−</mo><mi>j</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo class="MathClass-bin">−</mo><mn>1</mn></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>B</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">)</mo></mrow></math>$ when $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi></math>$ is transverse to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math>$ . This generalizes a similar formula of D Gottlieb in the case of simple coefficients.

Keywords

fibration, local trivial fibration, Poincaré duality, local coefficient system, (co)homology with local coefficients

Mathematical Subject Classification 2000

Primary: 55M20

Secondary: 55S35

References

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Publication

Received: 16 March 2005

Revised: 25 April 2006

Accepted: 14 August 2006

Published: 4 October 2006

Authors

Daciberg Lima Gonçalves
Dept. de Matemática IME - USP Caixa Postal 66.281 CEP 05311-970 São Paulo - SP Brasil

Peter Wong
Department of Mathematics Bates College Lewiston, ME 04240 USA

Volume 6, issue 3 (2006)