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Fundamental classes not representable by products. (English) Zbl 1168.53024

It is proved that rationally essential manifolds with suitable large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds. Large classes of examples including all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type are presented. It is also shown that non-positively curved closed manifolds and certain surface bundles over surfaces do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F67 Hyperbolic groups and nonpositively curved groups
57N65 Algebraic topology of manifolds
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