By Efton Park

ISBN-10: 0511388691

ISBN-13: 9780511388699

ISBN-10: 0521856345

ISBN-13: 9780521856348

Topological K-theory is a key device in topology, differential geometry and index concept, but this is often the 1st modern advent for graduate scholars new to the topic. No heritage in algebraic topology is believed; the reader desire simply have taken the normal first classes in genuine research, summary algebra, and point-set topology. The booklet starts off with an in depth dialogue of vector bundles and comparable algebraic notions, through the definition of K-theory and proofs of an important theorems within the topic, akin to the Bott periodicity theorem and the Thom isomorphism theorem. The multiplicative constitution of K-theory and the Adams operations also are mentioned and the ultimate bankruptcy information the development and computation of attribute periods. With each very important element of the subject lined, and routines on the finish of every bankruptcy, this can be the definitive booklet for a primary direction in topological K-theory.

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**Example text**

5 The class of all topological spaces with continuous functions as morphisms is a category. 6 The class of all compact Hausdorﬀ spaces with continuous functions as morphisms forms a category. If there is no possibility of confusion, we will usually only mention the objects of a category, with the morphisms being understood from context. So, for example, we will refer to the category of topological spaces, the category of abelian groups, etc. 7 Let A and B be categories. 9 A very brief introduction to category theory 45 (i) a function that assigns to each object A in A an object F (A) in B, and (ii) given objects A and A in A, a function that assigns to each morphism α ∈ Hom(A, A ) a morphism F (α) ∈ Hom(F (A), F (A )).

4 For each natural number m, consider S m = {(x1 , x2 , . . , xm+1 ) ∈ Rm+1 : x21 + x22 + · · · + x2m+1 = 1}, the m-dimensional sphere of radius 1 centered at the origin. Then ⎛ ⎞ x21 x1 x2 x1 x3 · · · x1 xm+1 ⎜ x2 x1 x22 x2 x3 · · · x2 xm+1 ⎟ ⎜ ⎟ ⎜ ⎟ .. .. .. ⎝ ⎠ . . . 2 xm+1 xm+1 x1 xm+1 x2 xm+1 x3 · · · is an idempotent over S m . 5 Consider S 1 as the unit circle in C and view the twotorus T2 as S 1 × [0, 1] with the usual identiﬁcation of the ends: (z, 0) ∼ (z, 1) for each z in S 1 . Set E(z, t) = Rot(t) diag(z, 1)Rot−1 (t).

Eσ(m) ) in M(n1 + n2 + · · · + nm , C). Proof Let U be the block matrix that for each i has the ni -by-ni identity matrix for its (σ −1 (i), i) entry and is 0 elsewhere. Then U is invertible and implements the desired similarity. 2 Let X be compact Hausdorﬀ. For any two elements of Idem(C(X)), choose representatives E in M(m, C(X)) and F in M(n, C(X)) and deﬁne [E] + [F] = [diag(E, F)]. Then Idem(C(X)) is an abelian monoid. Proof Because diag(SES−1 , TFT−1 ) = diag(S, T) diag(E, F) diag(S, T)−1 for all S in GL(m, C(X)) and T in GL(n, C(X)), we see that addition respects similarity classes.

### Complex Topological K-Theory by Efton Park

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