By Efton Park

Topological K-theory is a key instrument in topology, differential geometry and index thought, but this can be the 1st modern advent for graduate scholars new to the topic. No historical past in algebraic topology is believed; the reader want merely have taken the normal first classes in genuine research, summary algebra, and point-set topology. The e-book starts with a close dialogue of vector bundles and similar algebraic notions, through the definition of K-theory and proofs of crucial theorems within the topic, resembling 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 sessions. With each very important element of the subject coated, and workouts on the finish of every bankruptcy, this can be the definitive booklet for a primary direction in topological K-theory.

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The set of equivalence classes of ∼ is denoted M(C(X)). 11 is an example of a direct limit; we often indicate this by writing M(C(X)) = lim M(n, C(X)). −→ We will usually identify M(n, C(X)) with its image in M(C(X)). We can view an element of M(C(X)) as a countably inﬁnite matrix with entries in C(X) and all but ﬁnitely many entries equal to 0. 12 Let X be compact Hausdorﬀ. Then M(C(X)) is an algebra without unit under the operations of matrix addition and multiplication and scalar multiplication.

Idempotents E and F in M(n, C(X)) are algebraically equivalent if there exist elements A and B in M(n, C(X)) such that AB = E and BA = F. 48 Preliminaries (a) Show that A and B can be chosen so that A = EA = AF = EAF and B = FB = BE = FBE. (b) Prove that algebraic equivalence deﬁnes an equivalence relation ∼a on the idempotents in M(n, C(X)). (c) Prove that E ∼s F implies that E ∼a F. (d) If E, F in M(n, C(X)) are algebraically equivalent, prove that diag(E, 0n ) is similar to diag(F, 0n ) in M(2n, C(X)).

Therefore ψ(a1 )+ψ(b2 )+ψ(c) = ψ(a2 )+ψ(b1 )+ψ(c), and so ψ(a1 −a2 ) = ψ(a1 )−ψ(a2 ) = ψ(b1 )−ψ(b2 ) = ψ(b1 − b2 ). By construction, the map ψ is a group homomorphism and is unique because it is determined by ψ. 7 Vect(X) vs. Idem(C(X)) 31 a group G(A) and a monoid homomorphism i : A −→ G(A) with the properties listed in the statement of the theorem. The map i is a monoid homomorphism, so there exists a group homomorphism i : G(A) −→ G(A) such that ij = i. Next, because j : A −→ G(A) is a monoid homomorphism, there exists a group homomorphism j : G(A) −→ G(A) such that ji = j.