Download e-book for iPad: A geometric approach to homology theory by S. Buoncristiano

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 valuable notion is that of a 'mock bundle', that is the geometric cocycle of a basic cobordism idea, and the most new result's that any homology idea is a generalized bordism concept. The e-book will curiosity mathematicians operating in either piecewise linear and algebraic topology specifically homology idea because it reaches the frontiers of present learn within the subject. The publication is additionally appropriate to be used as a graduate direction in homology idea.

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J ,~' I 86 1, I 1f I I 87 j j 1 4. )) of (V, q)-manil'ulds off S(P) (where is regarded as a V-manifold by part 3 of the data). I bordism 1 WI WI' Then M x S(W1) is bordant to W by the V- bordism - (nbhd. of S(W1)). 1 aL ==O. A V-manifold has no second stratum. P is then called a dosed (V/U x M, q)-manifold. There is an obvious notion of V/U x M-manifold with boundary and hence we have geometric homology and cohomology theories Notation. j ~~ to W x {I} There are long exact sequences Form the product B x C(M) and attach it W==Wx to a V-manifold.

F ~[t], with a reduction to SO of the stable normal bundle I ignore them. ) which the reader can check is a generalisation efficient Sequence. Let n~O(,; 4. •V Exactly similar constructions deal only with the [M1], [M2]' ... ) ==0, i::; 1: ; 1 , is a free polynomial algebra moreover we can take index(M1) ==1 take M ==CP 1 2. )==O 1 jsubtract an appropriate number of copies of (CP )J. t, I • Now define theories 2 i ==1, 2, ... as follows: I 89 = J1 nSo * ( " . Zll[2 J2 = J1 /M , and inductively Ji 2 = Ji-l /M ..

Let S be a discrete set and let X = BPL x S Labelling. ,and f the projecti~ Then a connected (X, f)-manifold is just a mani- IIOldlabelled by an element of S. • S gives this theory products. See also the remarks at the end of the next section. 3. SINGULARITIES ,.. BPL E(~) Our treatment of singularities is similar to that worked out by ,Cookeand Sullivan (unpublished) or to be found in Stone [9]. The theory of (X, f)-mock bundles is set up in exactly the same way as the theory of bundles. In order to have products one needs in addition a commutative diagram Suppose we are given a class isomorphism).

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A geometric approach to homology theory by S. Buoncristiano

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