Abstract. Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism φ :and L, n > 1. A corollary is that each (n − 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 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 k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms, in particular there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M.

We study the question: which integers k can be realized as the degree of a map between two given closed (n−1)-connected 2n-manifolds? (2000):57R19, 55M25 Mathematics Subject Classification

This note studies relations between Spin bundles, over a decomplex of dimension < 9, and their first two Spin characteristic classes. In particular by taking Spin characteristic classes, it is proved that the stable classes of Spin bundles over a manifold M with dimension < 8 are in one to one correspondence with the pairs of cohomology classes (qx, q2) € H4(M; Z) x H%(M\ Z) satisfying (qx U?2 + <¡2) m°d 3 + t/3 U (qx mod 3) = 0, where U¡ e H*(M; Z3) is the indicated Wu-class of M. As an application a computation is made for K Spin(M), where M is an eight-dimensional manifold with understood cohomology rings over Z, Z2 , and Z3. This enabled him to define the Spin characteristic classes for the stable class of a Spin bundle £ over a topological space X by the formula Qi(Ç) = g*Qi£H*i(X;Z), _

ABSTRACT. In this note we present a simple approach to the Lefschetz number for the self-maps of Lie groups. As an application it is proved that for any map /: G