#### Volume 6, issue 5 (2006)

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On submanifolds in locally symmetric spaces of noncompact type

### Jean-François Lafont and Benjamin Schmidt

Algebraic & Geometric Topology 6 (2006) 2455–2472
 arXiv: math.GT/0607242
##### Abstract

Given a connected, compact, totally geodesic submanifold ${Y}^{m}$ of noncompact type inside a compact locally symmetric space of noncompact type ${X}^{n}$, we provide a sufficient condition that ensures that $\left[{Y}^{m}\right]\ne 0\in {H}_{m}\left({X}^{n};ℝ\right)$; in low dimensions, our condition is also necessary. We provide conditions under which there exist a tangential map of pairs from a finite cover $\left(\stackrel{̄}{X},\stackrel{̄}{Y}\right)$ to the nonnegatively curved duals $\left({X}_{u},{Y}_{u}\right)$.

##### Keywords
locally symmetric space, duality, tangential map, Matsushima's map
##### Mathematical Subject Classification 2000
Primary: 53C35
Secondary: 57T15, 55R37, 57R42, 57R45