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D. (y. T h e 1-parameter subgroup, 01, that satisfies &(O) = h is called the 1-parameter subgroup generated by h, and is denoted by a h . I n particular, the 1-parameter subgroup generated by 0 is the constant map t t-+ e. Example: Let C. be the multiplicative group of nonzero complex 2. The exponential map 33 numbers: C. = {x E C I z # O}. Then the corresponding Lie algebra is @, considered as a real vector space. T h e 1-parameter subgroup generated by a vector h E C is given by q , ( t ) = exp th.

Its integral is called the volume of Snand is given by + 2m+l 1 . 3 ... (2m 2 ,m+l - 1) nm, n = 2m, m 2 1 n=2m+1, m! m>O. 14. De Rham cohomology. Let M be an n-manifold. Then ( A ( M ) ,6) is a graded differential algebra; its cohomology is denoted by H ( M ) = C= :, HP(M)and is called the de Rham cohomology algebra of M . T h e homomorphism v*: A ( M ) +- A ( N ) determined by a smooth map induces a homomorphism v#: H ( M ) t H ( N ) . If dim H ( M ) < 00, then the pth Bettinumber, b, ,of Mis dim Hp(M).

Lie Groups 32 are both orbits of xh (use the left invariance of Hence (cf. Proposition X, sec. 15, volume I) a(s xh), and agree at t = 0. (t). (3) o (1): Same proof as (2) o (1). D. Proposition V: T o every vector h E T,(G) corresponds a unique I-parameter subgroup, a , such that i ( 0 ) = h. Proof: T h e uniqueness is immediate from Proposition IV. Now we prove existence. According to Proposition X, sec. 15, volume I, for some E > 0 there is an orbit 010 : (-6, 6) -+ G, for X , , satisfying ao(0) = e.

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A Hacker's Revenge


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