A functorial model for iterated Snaith splitting with applications to calculus of functors

Arone, Gregory; Kankaanrinta, Marja

doi:10.1090/fic/019/01

Cited by 16 publications

(33 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We now make use of the fact that s k * appears as a differential on the E 1 page of the homotopy spectral sequence which we know, thanks to [6], strongly converges to π * (S 1 ). This second quadrant spectral sequence has E 1 −s,t = π S t (L 1 (s)), and converges to π t−s (S 1 ), so that E ∞ −s,t = 0 unless (s, t) = (0, 1).…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

The Whitehead conjecture, the tower ofS¹conjecture, and Hecke algebras of type A

Kuhn

2014

Journal of Topology

View full text Add to dashboard Cite

Dedicated to the memory of my father Harold Kuhn (1925Kuhn ( -2014 and uncle Leon Henkin (1921-2006, mathematicians and lifelong advocates for social justice.Abstract. In the early 1980's the author proved G. W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990's, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd dimensional spheres.In this paper we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense.Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras.

show abstract

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…Given V ∈ V C , results by Arone and coauthors Mahowald [7], Dwyer [4], and Kankaanrinta [6] combine to show that there is a strongly convergent tower of principal fibrations under S R⊕V ,…”

Section: The Tower Of S 1 Conjecturementioning

confidence: 98%

The Whitehead conjecture, the tower ofS¹conjecture, and Hecke algebras of type A

Kuhn

2014

Journal of Topology

View full text Add to dashboard Cite

show abstract

“…Let's verify that the comparison map ( * ) is a weak equivalence. Our approach is to resolve the coaugmented cosimplicial diagram (8) of J N -algebras term-by-term with the finite tower {J n • JN (−)} n , where 1 ≤ n ≤ N ; we are motivated by the work in [1]. This leads to the commutative diagram in Alg O of the form We will show by induction up the finite tower that holim ∆ (#) n is a weak equivalence for each 1 ≤ n ≤ N .…”

Section: Nilpotent Structured Ring Spectramentioning

confidence: 99%

“…Our approach, which is somewhat indirect and was motivated by the work in [1], is to play off the homotopy completion and TQ-completion constructions against each other: more precisely, we resolve the cosimplicial TQ-resolution of an O-algebra X, whose homotopy limit is the TQ-completion of X, term-by-term via the homotopy completion tower, but with respect to the operad τ M−1 O characterizing M -nilpotent O-algebras (Remark 2.5). We use this to establish a nilpotent TQcompletion retract result (Theorem 2.12), and as a consequence, we establish the following nilpotent TQ-Whitehead theorem.…”

Section: Introductionmentioning

confidence: 99%

A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

Ching¹,

Harper²

2018

Tbilisi Math. J.

View full text Add to dashboard Cite

The aim of this short paper is to prove a TQ-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a TQ-homology analog for structured ring spectra of Dror's generalized Whitehead theorem for topological spaces; here TQ-homology is short for topological Quillen homology. We also prove retract theorems for the TQ-completion and homotopy completion of nilpotent structured ring spectra.

show abstract

“…For F D id, the Goodwillie tower converges on simply connected spaces to the identity; see Goodwillie [19]. On a connected space K , however, it converges to the Bousfield-Kan completion proved in [3] by Arone and Kankaanrinta: holim n P n .id/.K/ ' Z 1 K…”

Section: Introductionmentioning

confidence: 98%

Homotopy nilpotent groups

Biedermann

Dwyer

2010

Algebr. Geom. Topol.

View full text Add to dashboard Cite

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define homotopy n-nilpotent groups as homotopy algebras over certain simplicial algebraic theories. This notion interpolates between infinite loop spaces and loop spaces, but backwards. We study the relation to ordinary nilpotent groups. We prove that n-excisive functors of the form F factor over the category of homotopy n-nilpotent groups.55P47, 55U35; 18C10, 55P35

show abstract

A functorial model for iterated Snaith splitting with applications to calculus of functors

Cited by 16 publications

References 0 publications

The Whitehead conjecture, the tower ofS¹conjecture, and Hecke algebras of type A

The Whitehead conjecture, the tower ofS¹conjecture, and Hecke algebras of type A

A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

Homotopy nilpotent groups

Contact Info

Product

Resources

About

A functorial model for iterated Snaith splitting with applications to calculus of functors

Cited by 16 publications

References 0 publications

The Whitehead conjecture, the tower ofS1conjecture, and Hecke algebras of type A

The Whitehead conjecture, the tower ofS1conjecture, and Hecke algebras of type A

A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

Homotopy nilpotent groups

Contact Info

Product

Resources

About

The Whitehead conjecture, the tower ofS¹conjecture, and Hecke algebras of type A

The Whitehead conjecture, the tower ofS¹conjecture, and Hecke algebras of type A