DEGREES OF MAPPINGS OF MANIFOLDS.

摘要

If f:M(,1) (--->) N(,1) and g:M(,2) (--->) N(,2) are maps of 1-connected Poincare complexes both of degree d we show that there is a degree d map f g: M(,1) M(,2) (--->) N(,1) N(,2) of connected sums. This shows that there are manifolds, e.g., (S('2) x S('2)) (S('2) x S('2)) which admit maps of arbitrary degree but which are not products with any sphere.; It is known by various arguments that compact Riemann surfaces of genus > 1 have self-maps of degree -1, 0, +1 only. We generalize one approach to aspherical manifolds with Hopfian fundamental groups: If (chi)(M) (NOT=) 0 then only degrees -1, 0, +1 are possible.; We study nilmanifolds (Compact homogeneous spaces G/D where G is a 1-connected nilpotent Lie group and D is a discrete subgroup) and observe that fundamental groups of nilmanifolds are formal groups over certain subrings of the rational numbers. We can therefore construct associated Lie algebras generalizing constructions of earlier authors.; We also observe that if a polynomial function (rho):Z('n) x Z('n) (--->) Z('n) defines a group structure then the group is nilpotent.; We show that the set of homotopy equivalence classes of maps from one nilmanifold to another can be represented by a set of matrices which correspond to Lie algebra maps of associated Lie algebras. Using this representation we show that the diagram; ; H*(M; R) ('f*) H*(N; R); (TURNEQ) (TURNEQ); H(,Lie)( (M)) H(,Lie)( (N)); commutes where (M) and (N) are the associated real Lie algebras of M and N and the verticle maps are Nomizu's (1954) isomorphisms.; As a consequence, the degree of a map of nilmanifolds is the determinant of an associated Lie algebra map. This gives interesting restrictions on possible degrees of maps of nilmanifolds, e.g., we show the existence of a nilmanifold which has self maps of degrees 0, +1 only.; Finally we give a classification Theorem for 2-step nilmanifolds and study the possible degrees of self-maps of certain 2-step nilmanifolds.