By S. Buoncristiano

The aim of those notes is to offer a geometric remedy of generalized homology and cohomology theories. The valuable inspiration is that of a 'mock bundle', that is the geometric cocycle of a common cobordism idea, and the most new result's that any homology thought is a generalized bordism thought. The ebook will curiosity mathematicians operating in either piecewise linear and algebraic topology specially homology thought because it reaches the frontiers of present examine within the subject. The publication can also be compatible to be used as a graduate direction in homology idea.

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**Additional resources for A geometric approach to homology theory**

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M is a (p, n)-manifold, M x I has a where > SM has codimension p. M is bordant to a (p, n)-manifold Q, whose last stratum ° The proof of 2. 3 is left to the reader. There is an obvious notion of a singular (p, n)-manifold in a space and thus we have the bordism group n (X, A; n pl. The following proposition follows directly from proposition 2. 3, using the proof of II 3. 1. 4. {nn ( , ; p)} on the category of topological spaces. 2. 5. SM is Then is S, by a p. -SM and -S: assume that SR consists of a set of equally labelled components, with label, say, bP E BP; v(-SM) SR x = normal bundle of -8M in -M=MX pl.

I - BPL x n of (n-l)-polyhedra(closedunderpl Then a closed £ -manifold is a polyhedron M each of n whose links lies in £. A theory of manifolds-with-singularity n consists oC for each n = 0, 1, .. , which satisfies: n 1. each member of oC is a closed oC I-manifold n n2. e. the suspension of an (n-l)-link is an n-link) ofa class m XxX £ (2. 1) f xf n- BPL 3. e. the cone on an (n-l)-link is never an -link). where E9 is the map given by Whitney sum. Using diagram (2. 1) external Then an oC-manifoldwith boundary is a polyhedron whose links lie products can be defined by q~x71 = m 0 (q~ x q1))' Similarly cap products ither in oC or CoCn_l' Then the boundary consists of points whose n are defined with the corresponding bordism theory (maps of (X, f)-mani- inks lie in the latter class, and is itself a closed oC I-manifold.

1. The sum r' = (g' + ... 3(M)- 0. Now we are entitled to define + gt')' Therefore take a labelled copy V ® r' V x cone(g~ + ... Tn-1 by r'. 3 : of V and form the polyhedron W = (V ® r') x L(r', G') UM, where L(r', G') is the link generated by r' of G". We label -n (-' G") - -n n ' n-l (-'' G') in G' and the union is taken along·by + gP (see Fig. 14). W is a (G', n)-manifold and provides the required bordism between V ® g' and V ®g'. (2) Exactness at 0 (-; G) n (a) lJI*lJl* = is a (0, n)-manifold.