Higher cohomology operations and R–completion

Blanc, David; Sen, Debasis

doi:10.2140/agt.2018.18.247

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Higher cohomology operations and

R

$R$ –completion

David Blanc and Debasis Sen

Algebraic & Geometric Topology 18 (2018) 247–312

DOI: 10.2140/agt.2018.18.247

Abstract

Let $R$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>R</mi></math>$ be either $F_{p}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mspace width="0.3em" class="thinspace"/><mi>p</mi></mrow></msub></math>$ or a field of characteristic $0$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn></math>$ . For each $R$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>R</mi></math>$ –good topological space $Y$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Y</mi> <mspace width="0.3em" class="thinspace"/></math>$ , we define a collection of higher cohomology operations which, together with the cohomology algebra $H^{*} (Y; R)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>H</mi></mrow><mrow><mo class="MathClass-bin">∗</mo></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> <mo class="MathClass-punc">;</mo><mi>R</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ , suffice to determine $Y$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Y</mi> </math>$ up to $R$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>R</mi></math>$ –completion. We also provide a similar collection of higher cohomology operations which determine when two maps $f_{0}, f_{1} : Z \to Y$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo class="MathClass-punc">,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">:</mo> <mi>Z</mi> <mo class="MathClass-rel">→</mo> <mi>Y</mi> </math>$ between $R$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>R</mi></math>$ –good spaces (inducing the same algebraic homomorphism $H^{*} (Y; R) \to H^{*} (Z; R)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>H</mi></mrow><mrow><mo class="MathClass-bin">∗</mo></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> <mo class="MathClass-punc">;</mo><mi>R</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">→</mo> <msup><mrow><mi>H</mi></mrow><mrow><mo class="MathClass-bin">∗</mo></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>Z</mi><mo class="MathClass-punc">;</mo><mi>R</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ ) are $R$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>R</mi></math>$ –equivalent.

Keywords

higher cohomology operation, Toda bracket, cosimplicial resolution

Mathematical Subject Classification 2010

Primary: 55P60

Secondary: 55N99, 55P15, 55P20

References

Publication

Received: 5 September 2016

Revised: 4 May 2017

Accepted: 22 May 2017

Published: 10 January 2018

Authors

David Blanc
Department of Mathematics University of Haifa Haifa Israel

Debasis Sen
Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Kanpur India

Volume 18, issue 1 (2018)