Cohomology of B O ( n 1 ) × ⋯ × B O ( n m ) with local integer coefficients

Richard Lastovecki

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 1, page 21-32
  • ISSN: 0010-2628

Abstract

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Let 𝒵 be a set of all possible nonequivalent systems of local integer coefficients over the classifying space B O ( n 1 ) × ⋯ × B O ( n m ) . We introduce a cohomology ring ⨁ 𝒢 ∈ 𝒵 H * ( B O ( n 1 ) × ⋯ × B O ( n m ) ; 𝒢 ) , which has a structure of a ℤ ⊕ ( ℤ 2 ) m -graded ring, and describe it in terms of generators and relations. The cohomology ring with integer coefficients is contained as its subring. This result generalizes both the description of the cohomology with the nontrivial system of local integer coefficients of B O ( n ) in [Č] and the description of the cohomology with integer coefficients of B O ( n 1 ) × ⋯ × B O ( n m ) in [M].

How to cite

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Lastovecki, Richard. "Cohomology of $BO(n_1)\times \dots \times BO(n_m)$ with local integer coefficients." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 21-32. <http://eudml.org/doc/249525>.

@article{Lastovecki2005,
abstract = {Let $\mathcal \{Z\}$ be a set of all possible nonequivalent systems of local integer coefficients over the classifying space $BO(n_1)\times \dots \times BO(n_m)$. We introduce a cohomology ring $\bigoplus _\{\mathcal \{G\}\in \mathcal \{Z\}\} H^*(BO(n_1)\times \dots \times BO(n_m);\mathcal \{G\})$, which has a structure of a $\mathbb \{Z\}\oplus (\mathbb \{Z\}_2)^m$-graded ring, and describe it in terms of generators and relations. The cohomology ring with integer coefficients is contained as its subring. This result generalizes both the description of the cohomology with the nontrivial system of local integer coefficients of $BO(n)$ in [Č] and the description of the cohomology with integer coefficients of $BO(n_1)\times \dots \times BO(n_m)$ in [M].},
author = {Lastovecki, Richard},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {singular cohomology with local coefficients},
language = {eng},
number = {1},
pages = {21-32},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cohomology of $BO(n_1)\times \dots \times BO(n_m)$ with local integer coefficients},
url = {http://eudml.org/doc/249525},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Lastovecki, Richard
TI - Cohomology of $BO(n_1)\times \dots \times BO(n_m)$ with local integer coefficients
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 1
SP - 21
EP - 32
AB - Let $\mathcal {Z}$ be a set of all possible nonequivalent systems of local integer coefficients over the classifying space $BO(n_1)\times \dots \times BO(n_m)$. We introduce a cohomology ring $\bigoplus _{\mathcal {G}\in \mathcal {Z}} H^*(BO(n_1)\times \dots \times BO(n_m);\mathcal {G})$, which has a structure of a $\mathbb {Z}\oplus (\mathbb {Z}_2)^m$-graded ring, and describe it in terms of generators and relations. The cohomology ring with integer coefficients is contained as its subring. This result generalizes both the description of the cohomology with the nontrivial system of local integer coefficients of $BO(n)$ in [Č] and the description of the cohomology with integer coefficients of $BO(n_1)\times \dots \times BO(n_m)$ in [M].
LA - eng
KW - singular cohomology with local coefficients
UR - http://eudml.org/doc/249525
ER -

References

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  1. Brown E.H. Jr., The cohomology of B S O ( n ) and B O ( n ) with integer coefficients, Proc. Amer. Mat. Soc. 85 (1982), 283-288. (1982) MR0652459
  2. Čadek M., The cohomology of B O ( n ) with twisted integer coefficients, J. Math. Kyoto Univ. 39 2 (1999), 277-286. (1999) Zbl0946.55009MR1709293
  3. Feshbach M., The integral cohomology rings of the classifying spaces of O ( n ) and S O ( n ) , Indiana Univ. Math. J. 32 (1983), 511-516. (1983) Zbl0507.55014MR0703281
  4. Markl M., The integral cohomology rings of real infinite dimensional flag manifolds, Rend. Circ. Mat. Palermo, Suppl. 9 (1985), 157-164. (1985) Zbl0591.55007MR0853138
  5. Milnor J.W., Stasheff J.D., Characteristic Classes, Princeton University Press and University of Tokyo Press, Princeton, New Jersey, 1974. Zbl1079.57504MR0440554
  6. Spanier E., Algebraic Topology, McGraw-Hill, New York-Toronto, Ont.-London, 1966. Zbl0810.55001MR0210112
  7. Thomas E., On the cohomology of the real Grassman complexes and the characteristic classes of the n -plane bundle, Trans. Amer. Math. Soc. 96 (1960), 67-89. (1960) MR0121800

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