A simple universal property of Thom ring spectra

Camarena, Omar Antolín; Barthel, Tobias

doi:10.1112/topo.12084

Cited by 25 publications

(60 citation statements)

References 26 publications

(79 reference statements)

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“…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.…”

mentioning

confidence: 67%

“…(̄ , ) are equivalent. Hence, as described above, [3,Lemma 4.4] implies that ( ) ( ) ∕ ∕ 1 ≃ 1 0 ( (̄ )) ≃ ( + 1) ( ) .…”

mentioning

confidence: 81%

“…is called the "versal --algebra of characteristic " in [3] because it always admits a not-necessarily-unique map to any other -algebra on which multiplication by is nullhomotopic.…”

Section: The Effect Of Attaching a Structured Cellmentioning

confidence: 99%

“…This yields an 1 -map Ω 2 −1 → 1 ( ∕Ω 2 ( − 1)) ≃ 1 ( ( )). This is equivalent data to a map of pointed spaces 2 −2 → 1 ( ( )), and we take to be the element canonically associated to it by [3,Definition 4.8]. It should be remarked that very little is known about the homotopy groups of ( ) in general.…”

Section: Introductionmentioning

confidence: 99%

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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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mentioning

confidence: 67%

“…(̄ , ) are equivalent. Hence, as described above, [3,Lemma 4.4] implies that ( ) ( ) ∕ ∕ 1 ≃ 1 0 ( (̄ )) ≃ ( + 1) ( ) .…”

mentioning

confidence: 81%

“…is called the "versal --algebra of characteristic " in [3] because it always admits a not-necessarily-unique map to any other -algebra on which multiplication by is nullhomotopic.…”

Section: The Effect Of Attaching a Structured Cellmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…After further composing with the suspension spectrum functor

{normalΣ}^{\infty} : Spaces ⟶ Spectra,

the

J

‐homomorphism takes values in the subgroupoid of invertible spectra and their automorphisms, known as the Picard category of Spectra. This implies that

J

factors through the group completion of the category of complex vector spaces, giving a stable

J

functor

J : B U \times Z ≃ {Virtual Complex Vector Spaces}^{≃} ⟶ Pic false(Spectra false) \subset Spectra .

Since this stable

J

functor is symmetric monoidal, its homotopy colimit,

M U P

, acquires an

E_{\infty}

‐ring structure [3, 6, 15]. Convention Throughout this paper, whenever we refer to

M U P

as an

E_{\infty}

‐ring spectrum, we always give it the

E_{\infty}

‐ring structure constructed above.…”

Section: Introductionmentioning

confidence: 99%

Exotic multiplications on periodic complex bordism

Hahn

Yuan

2020

Journal of Topology

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Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of BU. This presents an E∞ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the E∞-ring MUP. Here, we prove that these two E∞-rings are in fact different, though the underlying E2-rings are equivalent. Nonetheless, we prove that both rings E∞-orient KU ∧ 2 and other forms of K-theory.

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Hermitian K-theory for stable $$\infty $$-categories I: Foundations

Calmès

Dotto

Harpaz

et al. 2022

Sel. Math. New Ser.

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This paper is the first in a series in which we offer a new framework for hermitian $${\text {K}}$$ K -theory in the realm of stable $$\infty $$ ∞ -categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré $$\infty $$ ∞ -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived $$\infty $$ ∞ -categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on $$\infty $$ ∞ -categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré $$\infty $$ ∞ -categories, showing in particular that they form a bicomplete, closed symmetric monoidal $$\infty $$ ∞ -category. We also study the process of tensoring and cotensoring a Poincaré $$\infty $$ ∞ -category over a finite simplicial complex, a construction featuring prominently in the definition of the $${\text {L}}$$ L - and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré $$\infty $$ ∞ -category using generators and relations. We extract its basic properties, relating it in particular to the 0th $${\text {L}}$$ L - and algebraic $${\text {K}}$$ K -groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

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A simple universal property of Thom ring spectra

Cited by 25 publications

References 26 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

Exotic multiplications on periodic complex bordism

Hermitian K-theory for stable $$\infty $$-categories I: Foundations

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