By William Fulton

This publication introduces the real rules of algebraic topology through emphasizing the relation of those principles with different components of arithmetic. instead of deciding on one viewpoint of contemporary topology (homotropy concept, axiomatic homology, or differential topology, say) the writer concentrates on concrete difficulties in areas with a number of dimensions, introducing purely as a lot algebraic equipment as beneficial for the issues encountered. This makes it attainable to determine a greater diversity of significant good points within the topic than is usual in introductory texts; it's also in concord with the historic improvement of the topic. The booklet is aimed toward scholars who don't inevitably intend on focusing on algebraic topology.

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Extra info for Algebraic Topology: A First Course (Graduate Texts in Mathematics, Volume 153)

Sample text

Contracting the indices k and i by setting k = i and summing over 1 ≤ i ≤ m, we have the Ricci identity f i ji − f ii j = fl Rl j . For p ≤ m, we will now take the convention on the indices so that 1 ≤ i, j, k, l ≤ m, 1 ≤ α, β, γ ≤ p, and 1 ≤ a, b, c, d ≤ p − 1. Let ω ∈ p (M) be an exterior p-form defined on M. Then in terms of the basis, we can write ω = ai1 ... i p ωi p ∧ · · · ∧ ωi1 , where the summation is being performed over the multi-index I = (i 1 , . . , i p ). With this understanding, we can write ω = aI ωI .

30 Geometric Analysis Hence, S(ω), ¯ ω¯ = Riα jα kl ai1 ... jα ... i p ωl ∧ ωk , ai1 ... jβ ... i p ω jβ ∧ ωiβ = Riα jα iβ jβ ai1 ... jα ... i p ai1 ... jβ ... i p = E(ω), ω . 5. 8 A vector field X = ai ei is a Killing vector field if ai, j + a j,i = 0 for all 1 ≤ i, j ≤ m. 9 The infinitesimal generator of a one-parameter family of isometries of M is a Killing vector field on M. Proof Let φt : M → M be a one-parameter family of isometries parametrized by t ∈ (− , ). If {e1 , . . , em } is an orthonormal frame at a point x ∈ M given by the normal coordinates centered at x, then the fact that φt are isometries implies that dφt (ei ), dφt (e j ) = ei , e j = δi j .

Iα ∧ . . ∧ ωi1 , and (−1)α+ p δω = 2 +1 ai1 ... iα ... i p ,i α ωi p ∧ . . ∧ ωˆ iα ∧ . . ∧ ωi1 . 1≤α≤ p Computing directly gives − ω = δ(ai1 ... i p , j ω j ∧ ωi p ∧ . . ∧ ωi1 ) + d[(−1)α+ p = (−1)α+( p+1) 2 +1 2 +1 ai1 ... i p ,i α ωi p ∧ . . ∧ ωˆ iα ∧ . . ∧ ωi1 ] ai1 ... i p , jiα ω j ∧ ωi p ∧ . . ∧ ωˆ iα ∧ . . ∧ ωi1 + (−1)( p+1)+( p+1) + (−1)α+ p 2 +1 2 +1 ai1 ... i p , j j ωi p ∧ . . ∧ ωi1 ai1 ... i p ,iα j ω j ∧ ωi p ∧ . . ∧ ωˆ iα ∧ . . ∧ ωi1 . 3) implies that ai1 ... i p ,iα j = ai1 ...