A geometric approach to homology theory by S. Buoncristiano

By S. Buoncristiano

The aim of those notes is to provide a geometric therapy of generalized homology and cohomology theories. The valuable inspiration is that of a 'mock bundle', that's the geometric cocycle of a normal cobordism concept, and the most new result's that any homology conception is a generalized bordism idea. The ebook will curiosity mathematicians operating in either piecewise linear and algebraic topology in particular homology idea because it reaches the frontiers of present study within the subject. The booklet is additionally appropriate to be used as a graduate direction in homology thought.

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There is an obvious notion of V/U x M-manifold with boundary and hence we have geometric homology and cohomology theories Notation. j ~~ to W x {I} There are long exact sequences Form the product B x C(M) and attach it W==Wx to a V-manifold. {oj X a ==O. If (W, S(W» is a V /U x M-manifold then S(W) x M bounds W - (nbhd. of S(W)). L X B. in W x I by the identity on S(W) x C(M). • Vq_l (X, A). V, S(W)) be a V/U x M-manifold such that S(W) L. bounds the V-manifold (V/U x M)* and (V/U x M If U ==V then we collapse the notation to V/M.

Omology theories Tt,(,), "-' T£*( ,). Moreover the proofs of the Poincare uality and Thorn isomorphism theorems are unaltered. ee below, that products are not defined in general (but cap product with I 84 I I 85 the fundami:mtalclass of ;, manifold (amalgamation) is :ways defined). the point. , Products (§4 contains details of killing. ) To combine £-theory with the restriction on the normal bundle :,'Of the last section, it is necessary to use the notion of 'normal block Suppose M and N are closed £-maJiitolds then M x N is in general not an £-manifold.

Sequence of nn (-; G) is pure. 10. For every pair S1(X,A; G) is more appropriate Pi)' to the any i-canonical resolution of G). 4. b* have been In the following we may use whichever of the equivalent functors Corollary 3. 9. integer n as and O*(X, A; G). They have different PG) establishes a natural equivalence gives us the following co~fic~nt 3. 1 p) -+ n*(-; p'). while the latter is natural on the category of abelian groups. We are now able to say something about the splitting of the unithe sequence is natural on the category T*: rl*(-; The above treatment of functoriality can be summarized follows.

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