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By S. Buoncristiano

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

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M is a closed V, n -manifold. theory can be made into 'coefficients Z ' by adding orientations and an i n j 3. There is a natural way of regarding W x M as a V-manifold, untwisted neighbourhood for the singularity (see Chapter ill, Example; .. ,oreach U-mamfold W (e. g. by relabelhng or forgettmg some structure). •. Then the theory VIU x M is defined by considering polyhedra P ting Z -homology (see §S). I .... f1tha two stage stratIfIcatIOn P::) S(P) and extra structure such that: j. 3. £0 = {p'l, £1 = {xix ~ SOJ, £n is all closed £n-l- manifolds.

Proof of exactness in W=_(SWXbP+lL(bP+l, 1. 2. ) p) '-------" MXI v(-S(M) Proof of Proposition 2. 5 (continued). Now let us look at the image of [SM] through the morphism _ I/> : P nn-p I8iF p id I8iI/>p nn-p I8iF p-l . i i C of Jt~e~oundar~ ~f ~he cdmPlement of a regular neighbourhood of V in the (n-p+ I)-stratum and a (p, n)-manifold n - p + 1. The singularities M' 8M have been resolved up to bardism. ] I8i})\J-I: this is nothing else than the bordism See Fig, 11. ] n n-p such that p+l ~P+l[SW] = [SM], Suppose first that SW is a set of components all labelled by bP+1 € BP+1, We can always reduce to the case M x II [SW] € (SW)kI8iiJ'+I and gets the desired manifold M'.

G' + g +, .. + g (g~ E G'; g. E G - G'). " + g, Therefore we attach a sheet q 1 (V x I) ® g' to W r' = g' +", ] along V and change the label r into a + g' + g', which is now a relation in G', p -- Let V ® r, V x (ii) component 'I' ®; r, V ® r, '" ... q,l r 2 / 1 r' ... r' l/~;h{" G-bordism Wand remove from it all the top dimensional strata which We show how to make W a G'-bordism by inserting new sheets. q,2 r N' To this purpose we reconsider the G'-manifold in general. - If [M] dl n(-; G), Then I/I(M) is a (G', n)-manifold Un(-; G').

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