By Efton Park

Topological K-theory is a key device in topology, differential geometry and index conception, but this can be the 1st modern creation for graduate scholars new to the topic. No heritage in algebraic topology is believed; the reader desire in simple terms have taken the traditional first classes in actual research, summary algebra, and point-set topology. The e-book starts with a close dialogue of vector bundles and similar algebraic notions, via the definition of K-theory and proofs of an important theorems within the topic, equivalent 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 info the development and computation of attribute periods. With each vital element of the subject lined, and workouts on the finish of every bankruptcy, this is often the definitive booklet for a primary direction in topological K-theory.

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**Example text**

14 Let X be a compact Hausdorﬀ space. Then the abelian monoids Idem(C(X)) and Vect(X) are isomorphic. Proof Deﬁne ψ : Idem(C(X)) −→ Vect(X) as ψ[E] = [Ran E]. 6. 13 yields that ψ is onto. We end this section by proving a result we will need in Chapter 3. 15 Let X be a compact Hausdorﬀ space, let V be a vector bundle over X, and suppose that W is a subbundle of V . Then there exists a vector subbundle W of V with the property that W ⊕ W is isomorphic to V . 12 implies that we can imbed V into a trivial bundle ΘN (X) for some natural number N .

X) = (x, x) for every x in X. Then V1 ⊕ V2 = Δ∗ (V1 V2 ). 5 Vector bundles 25 As in the case of the external Whitney sum, we can take the internal Whitney sum V1 ⊕ V2 ⊕ · · · ⊕ Vm of any ﬁnite number of vector bundles over X. 15 Let V be a vector bundle over a topological space X. The dual vector bundle of V is the set V ∗ of vector bundle homomorphisms from V to Θ1 (X). The dual vector bundle of V is a family of vector spaces over X because for each x in X, the elements of V ∗ restrict to linear maps from Vx to C.

Then for any abelian group H and monoid homomorphism ψ : A −→ H, there is a unique group homomorphism ψ : G(A) −→ H such that ψj = ψ. Moreover, G(A) is unique in the following sense: suppose that there exists an abelian group G(A) and a monoid homomorphism i : A −→ G(A) with the property that the image of i is a set of generators for G(A). Suppose further that for any abelian group H and monoid homomorphism ψ : A −→ H, there is a unique group homomorphism ψ : G(A) −→ H such that ψi = ψ. Then G(A) and G(A) are isomorphic.