The Homology of Iterated Loop Spaces

Cohen, Fred; Lada, Thomas J.; May, J. P.

doi:10.1007/bfb0080464

Cited by 376 publications

(603 citation statements)

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“…This follows from Proposition 2.3, and the fact that Dyer-Lashof operations commute with the evaluation [CLM,p.6,p.218].…”

Section: Cohomology Calculationsmentioning

confidence: 81%

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New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

Kuhn

2001

Cohomological Methods in Homotopy Theory

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Abstract. Let T (j) be the dual of the j th stable summand of Ω 2 S 3 (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T (j) → T (2j) → . . . is an infinite wedge of stable summands of K(V, 1)'s, where V denotes an elementary abelian 2 group. In particular, when one starts with T (1), one gets K(Z/2, 1) = RP ∞ as one of the summands.I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain finite spectra T (n, j) for n, j ≥ 0 (with T (1, j) = T (j)), dual to summands of Ω n+1 S N , conjecture generalizations of the above, and prove that these conjectures are correct in cohomology. So, for example, T (n, j) has unstable cohomology, and the cohomology of the hocolimit of a certain sequence T (n, j) → T (n, 2j) → . . . agrees with the cohomology of the wedge of stable summands of K(V, n)'s corresponding to the wedge occurring in the n = 1 case above.One can also map the T (n, j) to each other as n varies, and here the cohomological calculations imply a homotopical conclusion: the hocolimits that are nonzero, T (∞, 2 k ), for k ≥ 0, map to each other, giving rise to a fitration of HZ/2 which is equivalent to the mod 2 symmetric powers of spheres filtration.Our homotopical constructions use Hopf invariant methods and loopspace technology. These are quite general and should be of independent interest.To study the action of the Steenrod operations on the cohomology of our spectra, we derive a Nishida formula for how χ(Sq i ) acts on Dyer-Lashof operations. This should be of use in other settings.In an appendix, we explain connections with recent work by Greg Arone and Mark Mahowald on the Goodwillie tower of the identity.

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“…This follows from Proposition 2.3, and the fact that Dyer-Lashof operations commute with the evaluation [CLM,p.6,p.218].…”

Section: Cohomology Calculationsmentioning

confidence: 81%

“…The standard Nishida relations [CLM,p.6,p.214] tell us how (Sq r ) * commutes with Q s in H * (D n+1, * S −n ). Since χ(Sq r ) 6 acting on H * (T (n, * )) corresponds to (Sq r ) * acting on H − * (D n+1, * S −n ), we immediately have the following formula.…”

Section: New Nishida Relationsmentioning

confidence: 99%

New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

Kuhn

2001

Cohomological Methods in Homotopy Theory

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“…Fortunately, by the Adams-Priddy Theorem [6] BO® and BOig, are equivalent as infinite loop spaces (BO ss BSO since we are localized at an odd prime). May [20] describes some # polynomial generators explicitly as * polynomials when p = 2. Similarly, the infinite loop spaces BSB® and B5B® are equivalent.…”

Section: Examplesmentioning

confidence: 99%

Local 𝐻-maps of 𝐵𝑈 and applications to smoothing theory

Lance¹

1988

Trans. Amer. Math. Soc.

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ABSTRACT. When localized at an odd prime p, the classifying space PL/O for smoothing theory splits as an infinite loop space into the product CxN where C = Cokernel (J) and N is the fiber of a p-local H-map BU -> BU. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their fc-invariants to properties of the defining self-maps of BU. If Y is a smooth manifold, the set of homotopy classes [Y, N] is a certain subgroup of resmoothings of Y, and the ^-invariants of N generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of Y and frequently vanish. Introduction.When localized at an odd prime p, the classifying spaces for smoothings and for surgery theory can be written as products involving the p-local factors BO, C (the cokernel of J), and the fibers of certain B-maps BU -► BU. This paper studies the latter spaces, computing the cohomology of their Postnikov towers and relating the fc-invariants to the underlying geometry they classify. In smoothing theory there is an infinite loop space decomposition PL/O ss G x N where N is the fiber of a self-map of BU whose homotopy is the p-torsion in the groups of homotopy spheres bounding parallelizable manifolds. We show that for many smooth manifolds Y the resmoothings classified by N can be effectively computed and given a "characteristic variety" sort of description.Any B-map /: BU -► BU is determined up to homotopy by the characteristic sequence A = (Ai,A2,...), where /» is multiplication by Xj on Ti2j(BU) = Ztpy Because of the infinite loop space decomposition BU ss W x Q2W x ■■ ■ x U2p~4W [4,46] where n^W) = Zip) in dimensions t = 2j(p -1), we may write / as a product of B-maps with corresponding factorization of the fiber

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“…Segal [45] and Cohen [14] later proved that ~ is a group completion n in the general, non-connected, case. This means that w0f~nEnx is the universal if s ~ Im c for c ~ L.…”

Section: (L)~) ~ (L -{Y}x Il-{y}) If X(y) = "~ •mentioning

confidence: 99%

“…The crucial reason for the usefulness of the approximation theorem is that we have very good homological understanding of the filtered spaces C Xo n Historical background and complete calculations of H,~..~QX and H,~.,~CcoX (including X = S O , when the latter is ~ H~..BZq) are in [14,11. Cohen [14,111] has given q>_0 complete calculations of H,~n~,nx and H, CnX.…”

Section: (L)~) ~ (L -{Y}x Il-{y}) If X(y) = "~ •mentioning

confidence: 99%