We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree.
"Nonzero degree tangential maps between dual symmetric spaces." Algebr. Geom. Topol. 1 (2) 709 - 718, 2001. https://doi.org/10.2140/agt.2001.1.709