Nonzero degree tangential maps between dual symmetric spaces. (English) Zbl 1066.53100

The author studies the dual symmetric spaces. Let \(G\) be a real semisimple algebraic Lie group and K be its maximal compact subgroup. Let \(G_c\) be the complexification of \(G\) and \(G_u\) be a maximal compact subgroup of \(G_c\). The factor spaces \(G/K\) and \(X_u=G_u/K\) are called dual symmetric spaces of noncompact and compact type, respectively.
Okun constructs a tangential map from a locally symmetric space of noncompact type to its dual compact symmetric space. He proves that if Lie groups \(G_u\) and \(K\) have equal ranks then the induced map in cohomology is Matsushima’s type map [see Y. Matsushima, Osaka Math J. 14, 1–20 (1962; Zbl 0118.38401)]. Therefore the constructed map has a nonzero degree in the equal rank case.


53C35 Differential geometry of symmetric spaces
57T15 Homology and cohomology of homogeneous spaces of Lie groups
55R37 Maps between classifying spaces in algebraic topology
57R99 Differential topology


Zbl 0118.38401
Full Text: DOI arXiv EuDML EMIS


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