By Armand Borel, Edward Halpern

ISBN-10: 3540039074

ISBN-13: 9783540039075

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**Additional info for Topics in the Homology Theory of Fibre Bundles**

**Example text**

Of Polyt. Osaka City Univ. 2 (1952), lol-14o. [6] J. Leray, Sur la forme topologlque des espaces et sur les points fixes des repr~sentatlons, J. Math. Pures Applic. 54 (1945), 95-167. [7] J. , 1784-1786. [8] H. Samelson, Beltr~ge zur Topologie der Gruppen-Mannigfaltlgkelten, (1941), lo91-1137. 42 52 CHAPTER III UNIVERSAL BUNDLES AND CLASSIFYING 16. Universal bundles and classifying Let G be a compact Lie group. E(n,G) whose structural space B(n,G) spaces. An n-universal the following classification classes of principal correspondence bundle for G is a principal bundle group is G and whose first n homotopy groups vanish.

X m) is a simple (c) dim H = 2m. Hopf a l g e b r a over a perfect system for H, to show (c) + (a). ,s m be the r e s p e c t i v e Proposition s i = 2 for all i. 6. ,x m) be a heights. Then by (c) (a). If H is a finite d i m e n s i o n a l P(H,t) field odd, (a) ~ (b) § (c). 3) the are equivalent: (a) system of type (M). Then by system of generators. 5. If H is a finite d i m e n s i o n a l Kp, p ~ 2, t h e n the f o l l o w i n g Clearly of type Hopf a l g e b r a over a perfect has the form (i + t k2) ...

Is a of X. shall omit writing). We define (a,x) space E if there = (ax)-y, (ii) ae = a which together on a t o p o l o g i c a l § H,(E • X) a map | H~(X) PontrJagin § H~(E). product (thus a g e n e r a l i z e d and write "PontrJagin fashion to above we obtain a map r (a ~ x) = a v x. cap" product) as 20 $~: H~(E) For a r H*(E), u ~ H,(X) we define § H*(E) a v u r H~(E) by = , v ~ H ~ ( E ) , where the second v denotes the g e n e r a l i z e d r | H*(X). P o n t r J a g i n product.

### Topics in the Homology Theory of Fibre Bundles by Armand Borel, Edward Halpern

by John

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