By William S. Massey

ISBN-10: 038797430X

ISBN-13: 9780387974309

"This e-book is meant to function a textbook for a path in algebraic topology first and foremost graduate point. the most themes lined are the class of compact 2-manifolds, the basic staff, protecting areas, singular homology conception, and singular cohomology concept. those themes are constructed systematically, averting all pointless definitions, terminology, and technical equipment. at any place attainable, the geometric motivation in the back of a few of the options is emphasised. The textual content includes fabric from the 1st 5 chapters of the author's past e-book, ALGEBRAIC TOPOLOGY: AN advent (GTM 56), including just about all of the now out-of-print SINGULAR HOMOLOGY thought (GTM 70). the cloth from the sooner books has been rigorously revised, corrected, and taken as much as date."

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Additional resources for A Basic Course in Algebraic Topology (Graduate Texts in Mathematics)

Example text

P=1 Let us choose our linear map u to be the map u( u( j) (ek ) = δjk f . Then (u( j) )pm = δ p δjm , j) : E1 → E2 defined by and consequently, (Tu( j) )ki = 1 |G| (ρ2 (g))k (ρ1 (g −1 ))ji . 8. If ρ1 and ρ2 are inequivalent, then Tu( j) is always zero, whence (i). If E1 = E2 = E and ρ1 = ρ2 = ρ, then 1 |G| (ρ(g))k (ρ(g −1 ))ji = (Tu( j) )ki = g∈G Tr u( j) δki δ j δki = , dim E dim E which proves (ii). 10. Let (E1 , ρ1 ) and (E2 , ρ2 ) be unitary irreducible representations of G. We choose orthonormal bases in E1 and E2 .

20 Chapter 2 Representations of Finite Groups By the previous theorem, N (χρ | χρ ) = m2i . i=1 Hence we have the following result. 19 (Irreducibility Criterion). A representation ρ is irreducible if and only if (χρ | χρ ) = 1. 1 Definition In general, if a group G acts on a set M , then G acts linearly on the space F(M ) of functions on M taking values in C by (g, f ) ∈ G × F(M ) → g · f ∈ F(M ), where ∀x ∈ M, (g · f )(x) = f (g −1 x). We can see immediately that this gives us a representation of G on F(M ).

For any continuous f with compact support, we set μ(f ) = G f (g)dμ(g). Let h ∈ G and consider μh (f ) = G f (gh)dμ(g), that is, μh (f ) = μ(f ◦ rh ). Then ∀k ∈ G, μh (f ◦ lk ) = f (kgh)dμ(g) = G f (gh)dμ(g) = μh (f ), G and hence, by the uniqueness of left-invariant measures up to a factor, there is a positive real number Δ(h) depending on h satisfying μh (f ) = Δ(h)μ(f ). If G is compact, the constant function 1 is integrable. Therefore we obtain μh (1) = μ(1) = Δ(h)μ(1). , f (gh)dμ(g) = G f (g)dμ(g), G for every h ∈ G.

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A Basic Course in Algebraic Topology (Graduate Texts in Mathematics) by William S. Massey


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