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 principal proposal 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 conception. The booklet will curiosity mathematicians operating in either piecewise linear and algebraic topology in particular homology conception because it reaches the frontiers of present examine within the subject. The ebook can be appropriate to be used as a graduate path in homology idea.

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**Example text**

E. V ® w(bI) = aV. Take a copy aV consists of a number of copies of V (non constantly labelled in general); oM. Then, if oM has singularities up to codimension p and M s[W] ~ [V] + im (a) 2 O. > l l 0 n n- 1) [8M] + im ~ . It is straightforward the sequence Thus there is a well defined map: s : nn (P) Fig. 11 1m ~2' E E [W] we can assume, by Lemma 2. 8, 8W' in codimension at most one and that there W' - VI with singularities 8N in codimension two Then [sW] - [8W'] = ¢2[8W] and so s is an epimorphism.

He also proves that L[W] = then ) 1 for brevity of notation). CP where a. ) = 1 It follows that a. -+ L is as in diagram 2 = M1 , 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].

P ~hen -I/>p_1I/>p(bP) EKerl/>~_l Ofth dO " "t " and therefore it determines a basis element, b , in F Ker I/>p' 1; b'P has cyc 1es (M , f) , (M , f) are b or dan t 1 e 1Sl01l1umon 2 " (X A) B d" ° ° 1 1 t" a canonical word w(b'P) and cancellation rule c(b'P) induced from those (M1 U - M2'1 f 1 IUf 2 ) ~ or ds 111 , . p __ of 0- through the map (I/> lfi) Therefore the assignment in the set of singular (p, n)-cycles of (X, A). Denote the bordism class of p-l' p-£' tfl- (b'P " w(b'P) c(b'P)) defines lfip with the required properties.

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

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