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Journal of the London Mathematical Society Advance Access published online on February 19, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn089
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© 2009 London Mathematical Society

Fundamental classes not representable by products

D. Kotschick

Mathematisches Institut
Ludwig-Maximilians-Universität München
Theresienstr. 39
80333 München
Germany

C. Löh

Mathematisches Institut
Westfälische Wilhelms-Universität Münster
Einsteinstr. 62
48149 Münster
Germany
clara.loeh@uni-muenster.de

We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.

2000 Mathematics Subject Classification 53C23 (primary), 20F34, 20F67, 57N65 (secondary).

Received July 8, 2008;
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