By Werner Hildbert Greub, Stephen Halperin, Ray Vanstone

ISBN-10: 0123027020

ISBN-13: 9780123027023

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**Additional resources for Connections, Curvature, and Cohomology: 2 **

**Example text**

Assume that P is a representation of G in W such that P(x) d 0 = d 0 P(x), x E G. I. Lie Groups 42 Then P(x) determines a linear map P(x),: H( W )+ H( W ) and P,: x F+ P(x), is a representation of G in H ( W ) . On the other hand, the representation, P’, of E satisfies P’(h) o d = hEE d o P’(h), (differentiate the relation above). Hence P’(h) determines an operator P’(h), in H ( W ) and (P‘),: h i--t P’(h)# is a representation of E in H( W). 10. The adjoint representation. Each a E G determines the inner automorphism, r, , of G given by T,(x) x E G.

M T h e unique cohomology class u,,,E HE(M) such that ;J w,,, = 1 is called the orientation class. If M is compact, then H,(M) = H(M) and SO bp = b,, . T h e map @ @ Y I--, @ x Y (cf. sec. 13) defines homomorphisms K: A ( M )@ A ( N ) -+ A(M x N ) and K ~ A: c ( M )@ Ac(N)-+ Ac(M x N ) . These induce the Kiinneth homomorphisms K#: H ( M ) @ H ( N ) - + H ( M x N ) and H c ( M )@ H c ( N ) + Hc(M x N ) . (K~)#: ( K J # is always an isomorphism, while K # is an isomorphism if either H(M) or H(N) has finite dimension.

In view of Corollary I to Proposition VI, sec. 6, there is a neighbourhood V of 0 in T,(H) which exp,, maps diffeomorphically onto a neighbourhood U of e in H . Without loss of generality we may assume that a@)€ Define a continuous map u, It I < 1. /3: I -+ V ( I = {t E R I I t I < I}) by B ( t ) = expi' a(t). Since a is a homomorphism, Hence q P(t) E V if and only if Fix q # 0. Consider the set 0 E (I/dlI 4 * E v>* The above relation shows that this set is both closed and open in (1/q)I, and hence equal to (l/q)I.