# Algebraic and Classical Topology. The Mathematical Works of by John Henry Constantine Whitehead PDF

ISBN-10: 008009872X

ISBN-13: 9780080098722

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If B is a product bundle, then X(B) = 0, whence X(B) = 0. 11) in (9), it follows that E induces a homo­ morphism . 1\ u which is univalent if and only if E-i(0)cPiTn(S*). 4) _1 This will be the case, for example, if q = 3 or 7, for then £ (0) = 0, or iff n < 2q—2 (11). Since E oJ = —J o i* we have E*\(B) = —JX(B). 4) is satisfied. -P. Serre (12, p. 498) that 7rr(Sq) is finite except when r = q or when r = 2q— 1 and g is even. Since n > 1 it follows that An>(Z is finite if n ^ g. If n = g, then £ * is univalent and E* AniQ c 7r 2fl (^ +1 ), which is finite.

T Cf. 1) in (10). 1) HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES 45 Let n > 1 and, as in § 1, let pq: ir^RJ -+ 7rn_1(iJa) be the automorphism induced by the map r->pqrpqi where pqeOq is the reflection in the hyperplane tq_x = 0. 1) with T replaced by T. Let 0: F w x F f l - > F n x F « be defined by 0(y,z) = (y,pqz). Since Pq a* = a« and Pq T(y)uq z = T'{y)Pq uqz = T(y)uq Pq z it follows that pq o g = g' o 0. The map 0 is of degree — 1. 10 in (1) that pqij = £+Aa, for some a e 7rn(£«). 3) that (-t g ) o J£ = -Jpqi = -J£+Poc.

47 (1946), 779-85. 19. G. W. WHITEHEAD, A generalization of the Hopf invariant, Ann. , 51 (1950), 192-237. 20. G. W. WHITEHEAD, The (n -f 2)™* homotopy group of the n-sphere, Ann. , 52 (1950), 245-7. 21. G. W. WHITEHEAD, On the Freudenthal Theorems, Ann. , vol. 57 (1953), 209228. 22. J. H. C. WHITEHEAD, On adding relations to homotopy groups, Ann. , 42 (1941), 409-28. 23. J. H. C. WHITEHEAD, On the groups 7rr(F„, m) and sphere-bundles, Proc. London Math. Soc. (2), 48 (1944), 243-91. 24. J. H.