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By S. Buoncristiano

The aim of those notes is to offer a geometric therapy of generalized homology and cohomology theories. The primary concept is that of a 'mock bundle', that is the geometric cocycle of a normal cobordism conception, and the most new result's that any homology concept is a generalized bordism idea. The booklet will curiosity mathematicians operating in either piecewise linear and algebraic topology particularly homology thought because it reaches the frontiers of present study within the subject. The e-book can be compatible to be used as a graduate direction in homology concept.

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Example text

For general theories Hilton's 1 definition is equivalent to insisting that [each V-mock bundle T/. Remark 6. 1. Coefficients p is an example of killing, as des- cribed in §4. To make the notation fit with §4, let V = V ® F I (Va The trans- manifolds labelled by elements of B ) and U = V I8l F. • V is given by ignoring the label on the first 1 factor. 1 Then (V, p)-theory is the theory obtained from V simultaneously the elements {L(r, p)lr E B 1 1 by killing 1. I" 4(1), w:W~' ;t jOf L(b 2' p) - 81 is cobordant to zero for This is in fact sufficient to prove Theorem :ota::~d:::~ thenwe eaneompletethe eon,truetton plug in the bordism 1(inmost cases x T/ 2 D= D D of 81 to zero wherever appropriate and the construction coincides with the old one).

SM and -S: assume that SR consists of a set of equally labelled components, with label, say, bP E BP; v(-SM) SR x = normal bundle of -8M in -M=MX pl. L(~, v(-SM) {-I}; C M x {-I}; v(-SM) C SR x L(~, pl. a_t M 0 ~ -M', then _______ bordism R whose last stratum is still in codimension MUM' is a (p, n)-manifold with boundary isomorphic to g ---"__ --'0--__ Consider the following spaces: Proof. Cl(aM\M U aM'\M'). o g 0 M x 1', where l' = [0, -1]; 3. Let M be a (p, n)-manifold and X C M. Let N be a regular neighbourhood of X in M.

1. £ n is the class of (n-l)-spheres. £0 = sq-2 {p'), * The set of basic links {P' J. 2. Basic links are I £1 = {xix ~ SO or (n points) J, q 2: l j tioned above, this is ordinary bordism theory. here is As men- E £ and (n points) o (n points) E £. 1 J, £ = {xix ~ q Thus q-l S or 11, KILLING AND K-THEORY I In this section we give the general description of 'killing' an I lelementof a theory and apply it to give a geometric description of 1 fonnected K-theory at odd primes due essentially to Sullivan [ll].

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