Relative Thom spectra via operadic Kan extensions

Beardsley, Jonathan

doi:10.2140/agt.2017.17.1151

Cited by 11 publications

(20 citation statements)

References 19 publications

(36 reference statements)

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“…This paper extends the work of understanding the spectra ( ) begun in [5]. These spectra were introduced by Ravenel in [15] and were used in an essential way in Devinatz, Hopkins and Smith's proof of Ravenel's Nilpotence Conjecture [6].…”

Section: Introductionmentioning

confidence: 77%

A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra

Beardsley

2018

J. Homotopy Relat. Struct.

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We show that the homotopy groups of a connective -ring spectrum with an -cell attached along a class in degree are isomorphic to the homotopy groups of the cofiber of the self-map associated to through degree 2 . Using this, we prove that the 2 −1 homotopy groups of Ravenel's ( ) spectra are cyclic for all . This further implies that, after localizing at a prime, ( + 1) is homotopically unique as the 1 -( )-algebra with homotopy groups in degree 2 − 1 killed by an 1 -cell. Lastly, we prove analogous theorems for a sequence of -ring Thom spectra, for each odd , which are formally similar to Ravenel's ( ) spectra and whose colimit is also .-modules (̄ ) → ∕ ∕ which induces an isomorphism on homotopy groups in degrees less than 2 .Here, the spectrum ∕ ∕ is the versal -algebra on of characteristic , described for = ∞ in [20] and for all in [3]. It is constructed as by "attaching a -cell" to along in the category of --algebras. Often, versal algebras of this form are conceptually interesting but have difficult to determine homotopy groups, so the above theorem may be of interest beyond its uses in this paper.In Section 3 we apply the above theorem to prove that 2 −1 ( ( )) is cyclic, and that, moreover, it is generated by a certain element which was identified in [5]. This is Corollary 2:Corollary. The element

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Section: Introductionmentioning

confidence: 77%

A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra

Beardsley

2018

J. Homotopy Relat. Struct.

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“…To construct

H Z

as a Thom spectrum, we will combine the construction of

H {double-struckF}_{p}

as a Thom spectrum given in Theorem with the idea of intermediate Thom spectra from . Consider the fiber sequence

S^{3} false⟨ 3 false⟩ \overset{}{\to} S^{3} \overset{}{\to} K false(double-struckZ, 3 false)

which realizes the bottom of the Whitehead tower for

S^{3}

.…”

Section: Applicationsmentioning

confidence: 99%

“…Proof Following ideas of Beardsley , we will first construct an intermediate Thom spectrum

M ϕ_{p}

associated to a map

ϕ_{p} : S^{1} \to B G L_{1} false(M g_{p} false)

from the base of the fiber sequence . The properties of this spectrum allow us then to determine the homotopy groups of

M g_{p}

, from which the claim will follow.…”

Section: Applicationsmentioning

confidence: 99%

“…Let

Φ_{p}

be the operadic Kan extension of

F_{p}

along

π

with respect to the

E_{2}

‐operad. By ,

Φ_{p}

factors canonically through a map

ϕ_{p} : S^{1} \to B G L_{1} false(M g_{p} false)

, so we obtain the following (non‐commutative) diagram

We will now identify the Thom spectrum of

ϕ_{p}

in two different ways. On the one hand, by , there is an equivalence

M ϕ_{p} ≃ M f_{p}

E_{1}

‐ring spectra, which in turn is equivalent to

H {double-struckF}_{p}

by Theorem .…”

Section: Applicationsmentioning

confidence: 99%

“…By ,

Φ_{p}

factors canonically through a map

ϕ_{p} : S^{1} \to B G L_{1} false(M g_{p} false)

, so we obtain the following (non‐commutative) diagram

We will now identify the Thom spectrum of

ϕ_{p}

in two different ways. On the one hand, by , there is an equivalence

M ϕ_{p} ≃ M f_{p}

E_{1}

‐ring spectra, which in turn is equivalent to

H {double-struckF}_{p}

by Theorem . On the other hand, Theorem shows that

M ϕ_{p}

is also the versal

E_{0}

‐

M g_{p}

‐algebra of characteristic

χ_{p}

, where

χ_{p} \in π_{0} {(M g_{p})}^{\times}

is the element corresponding to

ϕ_{p}

.…”

Section: Applicationsmentioning

confidence: 99%

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A simple universal property of Thom ring spectra

Camarena

Barthel

2018

Journal of Topology

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We give a simple universal property of the multiplicative structure on the Thom spectrum of an n‐fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax scriptO‐monoidal functor. This allows us to relate Thom spectra to En‐algebras of a given characteristic in the sense of Szymik. As applications, we recover the Hopkins–Mahowald theorem realizing Hdouble-struckFp and HZ as Thom spectra, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.

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The enriched Grothendieck construction

Beardsley¹,

Wong²

2019

Advances in Mathematics

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We define and study opfibrations of V-enriched categories when V is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category B, there is an equivalence of 2categories between V-enriched opfibrations over the free V-category on B, and pseudofunctors from B to the 2-category of V-categories. This generalizes the classical (Set-enriched) Grothendieck correspondence.

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Relative Thom spectra via operadic Kan extensions

Cited by 11 publications

References 19 publications

A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra

A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra

A simple universal property of Thom ring spectra

The enriched Grothendieck construction

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