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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.

MSC:

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

Citations:

Zbl 0118.38401
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References:

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