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    Non-zero degree maps between 2n2n-manifolds

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    Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism Ο•:Hn(L;Z)β†’Hn(M;Z)\phi : H^n(L;Z)\to H^n(M;Z) can be realized by a map f:Mβ†’Lf:M\to L of degree kk for closed (nβˆ’1)(n-1)-connected 2n2n-manifolds MM and LL, n>1n>1. A corollary is that each (nβˆ’1)(n-1)-connected 2n2n-manifold admits selfmaps of degree larger than 1, n>1n>1. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree kk map from a closed orientable 4-manifold MM to a closed simply connected 4-manifold LL in terms of their intersection forms, in particular there is a map f:Mβ†’Lf:M\to L of degree 1 if and only if the intersection form of LL is isomorphic to a direct summand of that of MM.Comment: 18 page
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