A Remark on Maps Between Classifying Spaces of Compact Lie Groups

1988 ◽  
Vol 31 (4) ◽  
pp. 452-458 ◽  
Author(s):  
Zdzisław Wojtkowiak
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Finite Groups ◽  
Classifying Spaces ◽  
Compact Lie Groups ◽  
Rational Cohomology

AbstractWe show that two maps between classifying spaces of compact, connected Lie groups are homotopic after inverting the order of the Weyl group of the source if and only if they induce the same maps on rational cohomology. We shall also give some results on maps from classifying spaces of finite groups to classifying spaces of compact Lie groups. Among other things we construct a map from B(Z/2 + Z/2 4- Z/3) into BSO(3) which is not induced by a homomorphism.

Maps between p-completed classifying spaces II

1991 ◽  
Vol 118 (1-2) ◽  
pp. 133-137 ◽  
Author(s):  
Zdzisław Wojtkowiak
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Classifying Spaces ◽  
Compact Lie Groups ◽  
Homotopy Classes ◽  
Admissible Maps

We investigate maps between p-completed classifying spaces of compact connected Lie groups. Let G and G′ be two connected compact Lie groups. For a space X, let Xp be a p-completion of X. If p does not divide the order of the Weyl group of G, we give descriptions of the set of homotopy classes [(BG)p, (BG′)p] in terms of K-theory and in terms of “admissible” maps of Adams and Mahmud.

Maps between p-completed classifying spaces

1989 ◽  
Vol 112 (3-4) ◽  
pp. 231-235 ◽  
Author(s):  
J. Frank Adams ◽  
Zdzisław Wojtkowiak
Keyword(s):  
Lie Groups ◽  
Classifying Spaces ◽  
Compact Lie Groups ◽  
Complete Case ◽  
Topological Map ◽  
Maximal Tori ◽  
Local Case ◽  
Admissible Maps

SynopsisLet G and G' be two connected compact Lie groups with maximal tori T and T'. For a space X, let Xp be the p-completion of X. We will associate to each topological map f:(BG)p→(BG')p an “admissible map” ϕ:π1(T)⊗zZp→π1(T′)⊗zZp. We then show that the study of “admissible maps” in the p-complete case may be reduced to their study in the p-local case.

scholarly journals Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 21
Author(s):  
Ming Liu ◽  
Xia Zhang
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Lie Group ◽  
Weyl Groups ◽  
Module Structure ◽  
Compact Lie Groups ◽  
Compact Torus ◽  
Nonabelian Cohomology ◽  
Connected Lie Group

Given a compact connected Lie group G with an S 1 -module structure and a maximal compact torus T of G S 1 , we define twisted Weyl group W ( G , S 1 , T ) of G associated to S 1 -module and show that two elements of T are δ -conjugate if and only if they are in one W ( G , S 1 , T ) -orbit. Based on this, we prove that the natural map W ( G , S 1 , T ) \ H 1 ( S 1 , T ) → H 1 ( S 1 , G ) is bijective, which reduces the calculation for the nonabelian cohomology H 1 ( S 1 , G ) .

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