By S. Buoncristiano

ISBN-10: 0521209404

ISBN-13: 9780521209403

The aim of those notes is to provide a geometric remedy of generalized homology and cohomology theories. The crucial suggestion is that of a 'mock bundle', that is the geometric cocycle of a normal cobordism concept, 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 specially 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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**Example text**

3. • G' is a homomorphism of abelian groups, is a t. 1. resolution of G, y' is the canonical resolution which (a) M is a submanifold of oW (b) FIM =f and F(OW\M) C lfi extends to a linked map ~: p - A (c) M has at most three intrinsic strata (labelled by elements of BO, BI, B2). y' of G'; in a canonical way. = {F , I/>}, y' = {r', lfi' L We proceed by p p p P induction on p. Write (~) = (~o' ~I' ~/ For p = 0, put ~o(bo)=I/>E(bo) Proof. ° Let ° p ~ Define -(M, f) = (-M, f). Two singular (p, n)- for each b E B.

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'.

However, 2 1 1 0, and we can bord W to p' in the 2 Remark 4. 8. 1 1 = M W.. 2 1 There is a similar geometric description of con- nected KU-theory given by a similar construction using complex bordism s n~O ® Zr~] O. , where W. acts on Z[t ][t] by ®1 = is the coefficient of (M )n/4, since all the other M. ) ~ Z[t][t]. We will construct a natural transformation and the Conner-Floyd map [3]. 1/1 in the commutative 5. 1. Some theories which represent ---------~-- Let p be prime. W. in nS*O(,; Z[~]) where a..

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

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