## 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

Zbl 0118.38401
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### References:

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