Product rigidity (Ex)

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Let G be a group satisfying the Full Farrell-Jones Conjecture. Let M be a closed aspherical manifold with fundamental group G. Suppose that G \cong G_1 \times G_2, where the cohomological dimension of both G_1 and G_2 is different from 3 and 4. Show that M is homeomorphic to a product M_1 \times M_2 of two closed aspherical manifolds with \pi_1(M_i) \cong G_i for i = 1,2.

Hint: Use the fact that G_1 and G_2 are Poincaré duality groups, if G is a Poincar\'e duality group.

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