By Yu. B. Rudyak
For a few years, algebraic topology rested on 3 legs: "ordinary" cohomology, K-theory, and cobordism. This e-book is the 1st advisor within the topic of cobordism for the reason that R. Stong's encyclopaedic and influential notes of a iteration ago.It concentrates on Thom areas (spectra), orientability conception and (co)bordism idea (including (co)bordism with singularities and, specifically, Morava K-theories), framed by means of (co)homology theories and spectra.The writer has additionally played a carrier to the historical past of technology during this booklet, giving distinctive attributions. This similar care makes the e-book effortless to take advantage of by means of the scholar, for whilst proofs aren't given, particular references are.
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Extra info for On Thom spectra, orientability, and cobordism
Example text
K En+k −→ . . §1. Preliminaries on Spectra 43 Define ψk to be the composition Ωk εn+k α ψk : Ωk En+k − → Ωk En+k −−−−−→ Ωk+1 En+k+1 = Ω(Ωk En+k+1 ) Ωβ −−→ Ω(Ωk En+k+1 ). Then we have a homotopy commutative diagram En ⏐ ⏐ε ϕ0 −−−−→ n ΩEn+1 ⏐ ⏐ψ −−−−→ . . −−−−→ 1 Ωk En+k ⏐ ⏐ψ ϕk −−−−→ . . k Ωϕk Ωϕ0 ΩEn+1 −−−−→ Ω(ΩEn+2 ) −−−−→ . . −−−−→ Ω(Ωk En+k+1 ) −−−−→ . . where ψn ∼ = (Ωβ)◦α◦ϕn . Passing to telescopes, we get an obvious map ω : Tn → ΩTn+1 induced by the ψk ’s. Since every compact set in Tn is contained in some finite union m Ωk En+k × [k, k + 1], k=0 k lim we conclude that πi (Tn ) = − → πi (Ω En+k ) = πi−n (E).
Then f = g. (ii) Let E , E be cofinal in E, and let f : E → F and f : E → F be two equivalent maps. Then f |E ∩ E = f |E ∩ E . (iii) Every morphism contains a greatest element with respect to the above partial ordering. Proof. (i) Let {e, Se, . . , S k e, . . } be a cell of E where e is the cell of En . Since fn+k |S k e = gn+k |S k e for some k, we have fn |e = gn |e. (ii) This follows from (i). (iii) Fix a morphism ϕ. If (f , E ) ∈ ϕ and (f , E ) ≤ (f , E ), then (f , E ) ∈ ϕ. Hence, by Zorn’s Lemma, ϕ has a maximal element.
Furthermore, if f, g : S n → kX are two maps and H : S n × I → X is a pointed homotopy between kf and kg then w−1 H is a pointed homotopy between f and g, and so w∗ is injective. 50. Theorem. Let X be an arbitrary topological space having the homotopy type of a CW -space. Then w : kX → X is a homotopy equivalence. In particular, if X ∈ W and X has the homotopy type of a CW -space then ΩX has the homotopy type of a CW -space. 32 Chapter I. Notation, Conventions and Other Preliminaries Proof. Let Y be a CW -space, let f : Y → X be a homotopy equivalence, and let g : Y → X be homotopy inverse to f .