By S. Buoncristiano

ISBN-10: 0521209404

ISBN-13: 9780521209403

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

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16 (4) Exactness at 0n(-; G') q,*f3 (a) singularities = O. g~'. g. and lJIg. = g~. We construct a G-bordism Fig. 15 I I I I W: (3(M")- yf as follows. (i) (3) Exactness at 0n(-; G") (a) n {3lJ1* = O. Let M r be a G-manifold. I - (ii) M by changing each g~ into g. and r" into I I g'. We attach a new sheet (V x I) ® g' to the relabelled the singularities V ® r. M" along See Fig. 17 set of components of S(M). But V ® 0 - ¢ by a trivial G' - bordism. (b) Ker {3C 1m lJI*. Let M"n be a G"-manifold. sake of simplicity let us assume that the singularities For the of M" have only one component V ® r"; r" = L.

X, A) = (point, ¢). ----- M is a subset M °C M together with an embedding f : M' '+ M (of p-manifolds) such that f(M') = M. ° obtained from If M is a (p, n)-manifold, M by reversing (p, n)-manifolds have the following properties. 3. 1. of M with I, it is clear that 2. It involves a resolution of singularities. (p, n)-manifold. 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.

Is well-defined ----- = lj/ is defined on genera- € £, , (p) a € £, , (p) * (p) 1 Z -homology. Define a theory of singularities € £, , ••. 2 where p ---- by the basic (p) is a set with p 91 points in it. call an £-manifold a po-polyhedron. This is a ring WI:- theory (see §3), the ring closure of 'twisted Z -bordism' p £ (Example £ a n = {p'l, £ = {p Ip 1 = {(q)/q even is a closed £ n- l and inductively I-manifold with even Euler 3. 1(2)). An orientation for an n-dimensional p-polyhedron is a generator of H (P; Z ) ~ Z.

### A geometric approach to homology theory by S. Buoncristiano

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