By S. Buoncristiano

The aim of those notes is to offer a geometric therapy of generalized homology and cohomology theories. The significant suggestion is that of a 'mock bundle', that is the geometric cocycle of a normal cobordism conception, and the most new result's that any homology concept is a generalized bordism concept. The ebook will curiosity mathematicians operating in either piecewise linear and algebraic topology specifically homology idea because it reaches the frontiers of present examine within the subject. The publication is usually appropriate to be used as a graduate path in homology conception.

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Then a (p "p', p) 11' (p ® p')- around SM x SM'. dimensional stratum of L" is isomorphic to L(r, w(r ® r'). } g @ 3 r' g I8l 3 g' 3 3 Fig. 13 68 g' 69 We can now define a homomorphism x p, p' : n (-' *' p) ® n (-' *' 5. THE BOCKSTEIN SEQUENCE p') - n (-' * ' p ® p') Theorem 5. 1. On the category of short exact sequences of abelian groups by o- there is a natural connecting homomorphism x p, p' is of degree zero. Let p3 '" ljJ G' - G - G" - 0 be a 3-canonical resolution of _ G ® G'. - Then, by the proof of 3.

By assumption there is a G'-bordism I I I I We construct a G-manifold Mn as follows. (i) Since {3(M") has no singularities singularities W' : (3(M")-~, we can assume that W' has in codimension one at most, because otherwise we solve Fig. 17 the codimension-two stratum as in the proof of the universal-coefficient theorem. 74 75 (b) a G-bordism, Ker q,* c 1m p, Let M,n be a G'-manifold and W: M,n - From W we get a G"-manifold W" of dimension ¢ (n + 1) 6. as follows, (i) The proof of exactness is now complete and naturality is clear.

In particular there are functors bundles and Poincare duality for oriented manifolds) - more general argu- category of abelian groups and there is a universal coefficient sequence ments will in fact be given in §2. This theory will be denoted n*(,) place of ordinary manifolds. UK is a polyhedron each E (Ji K, p-l(a) More precisely, E(~) with projection by using p-manifolds in p) a (p, q)-mock bundle p: E(~) -+ K such that, for is a (p, q+i)-manifold with boundary p-l(&), called the block over a and denoted ~(a), and such that E(~(T), ~(a)) = E(T, a) n for each T < a E K.