By S. Buoncristiano

The aim of those notes is to provide a geometric remedy of generalized homology and cohomology theories. The valuable thought is that of a 'mock bundle', that's the geometric cocycle of a basic cobordism conception, and the most new result's that any homology concept is a generalized bordism idea. The e-book will curiosity mathematicians operating in either piecewise linear and algebraic topology specifically homology conception because it reaches the frontiers of present examine within the subject. The booklet is additionally appropriate to be used as a graduate path in homology thought.

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Defined as in the non-truncated objects are t. 1. resolutions A linked map between t. 1. resolutions case and there is a category, e, {n*(x, A; p)} on CP x ()) (G = category of topological pairs). from p by choosing a based kernel of lfi. 2 (X, A) is a pair 1 is the obvious map; r = FBI 1 iJ f in terms of the elements of G 1 (X, A), r Fix a p' E e obtained 2 ~ 2 M and a map r }; = a 2 (h), expressing A singular (P, n)-cycle in C ° f and lfil = 0I~I = FB2; B2 = {h; w(h), c(h) Ih E Ker lfi , w(h) is a word ~I(f, w(f)) which is a functor (M, f) consisting of a (p', n)-manifold 1 1 are the free abelian groups on G, Ker E, lfi B = {(f, w(f)) If E Ker E C F Ker E and w(f) is a word expressing and whose morphisms are linked maps.

P pup P Both these modifications are examples of restriction on the normal 6. COEFFICIENTS IN A GEOMETRIC THEORY block bundle system. For more information on the algebra behind pIn this section V denotes a general geometric theory, that is to polyhedra see Bullett [1]. codimension l' is the wreath product say, a theory with singularities, Example 5. 2. Euler spaces. This theory was invented by Akin and Sullivan [16] and has interesting properties. 92 Define link classes by labellings and generalised orientations, as in §§2, 3.

Sullivan. The transversality class and linking cycles in surgery theory. (1974), IV· Geometric theories characteristic Ann. of Math. 99 463-544. In this chapter we extend the notion of a geometric homology and cohomology (mock bundle) theory by allowing (1) singularities (2) labellings (3) restrictions on normal bundles. The final notion of a 'geometric theory' is in fact sufficiently general to include all theories (this being the main result of Chapter Vn). A further extension, to equivariant theories, will be covered in Chapter V.