Thom Spectra Research Articles

A chromatic vanishing result for TR

In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that T ( k ) T(k) -local TR vanishes on connective L n p , f L_n^{p,f} -acyclic E 1 \mathbb {E}_1 -rings for every 1 ≤ k ≤ n 1 \leq k \leq n and deduce consequences for connective Morava K-theory and the Thom spectra y ( n ) y(n) . The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive ∞ \infty -categories which was recently established by Córdova Fedeli [Topological Hochschild homology of adic rings, Ph.D. thesis, University of Copenhagen, 2023].

Fibrant resolutions for motivic Thom spectra

Fibrant resolutions for motivic Thom spectra

Perfectoid rings as Thom spectra

Perfectoid rings as Thom spectra

Thom spectra, higher THH and tensors in ∞–categories

Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space of an $E_\infty$-ring spectrum $R$. We determine the tensor $X\otimes_R Mf$ of the Thom $E_\infty$-$R$-algebra $Mf$ with a space $X$; when $X$ is the circle, the tensor with $X$ is topological Hochschild homology over $R$. We use the theory of localizations of $\infty$-categories as a technical tool: we contribute to this theory an $\infty$-categorical analogue of Day's reflection theorem about closed symmetric monoidal structures on localizations, and we prove that for a smashing localization $L$ of the $\infty$-category of presentable $\infty$-categories, the free $L$-local presentable $\infty$-category on a small simplicial set $K$ is given by presheaves on $K$ valued on the $L$-localization of the $\infty$-category of spaces. If $X$ is a pointed space, a map $g: A\to B$ of $E_\infty$-ring spectra satisfies $X$-base change if $X\otimes B$ is the pushout of $A\to X\otimes A$ along $g$. Building on a result of Mathew, we prove that if $g$ is \'etale then it satisfies $X$-base change provided $X$ is connected. We also prove that $g$ satisfies $X$-base change provided the multiplication map of $B$ is an equivalence. Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of $S^0$-base change.

Eilenberg Mac Lane spectra as p$p$‐cyclonic Thom spectra

Hopkins and Mahowald gave a simple description of the mod p $p$ Eilenberg Mac Lane spectrum F p ${\mathbb {F}}_p$ as the free E 2 $\mathbb {E}_2$ -algebra with an equivalence of p $p$ and 0. We show for each faithful 2-dimensional representation λ $\lambda$ of a p $p$ -group G $G$ that the G $G$ -equivariant Eilenberg Mac Lane spectrum F ̲ p $\underline{\mathbb {F}}_p$ is the free E λ $\mathbb {E}_{\lambda }$ -algebra with an equivalence of p $p$ and 0. This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral 2-subgroups of O ( 2 ) $\textnormal {O}(2)$ . The main new idea is that F ̲ p $\underline{\mathbb {F}}_p$ has a simple description as a p $p$ -cyclonic module over THH ( F p ) $\mathop {\rm THH}\nolimits (\mathbb {F}_p)$ . We show that our result is the best possible one in that it gives all groups G $G$ and representations V $V$ such that F ̲ p $\underline{\mathbb {F}}_p$ is the free E V $\mathbb {E}_V$ -algebra with an equivalence of p $p$ and 0.

Splittings of global Mackey functors and regularity of equivariant Euler classes

We establish natural splittings for the values of global Mackey functors at orthogonal, unitary and symplectic groups. In particular, the restriction homomorphisms between the orthogonal, unitary and symplectic groups of adjacent dimensions are naturally split epimorphisms. The interest in the splitting comes from equivariant stable homotopy theory. The equivariant stable homotopy groups of every global spectrum form a global Mackey functor, so the splittings imply that certain long exact homotopy group sequences separate into short exact sequences. For the real and complex global Thom spectra $\mathbf{MO}$ and $\mathbf{MU}$, the splittings imply the regularity of various Euler classes related to the tautological representations of $O(n)$ and $U(n)$.

Computational tools for twisted topological Hochschild homology of equivariant spectra

Twisted topological Hochschild homology of Cn-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum MUR. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson.

Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence ofPin⁡(2)\operatorname {Pin}(2)-equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing thePin⁡(2)\operatorname {Pin}(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem.We prove our theorem by analyzing maps between certain finite spectra arising fromBPin⁡(2)B\operatorname {Pin}(2)and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and thejj-based Atiyah–Hirzebruch spectral sequence.

The homotopy type of the topological cobordism category

We define a cobordism category of topological manifolds and prove that if d\neq 4 its classifying space is weakly equivalent to \Omega^{\infty -1}MT\mathrm{Top}(d) , where MT\mathrm{Top}(d) is the Thom spectrum of the inverse of the canonical bundle over B\mathrm{Top}(d) . We also give versions for manifolds with tangential structures and/or boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius–Madsen–Tillman–Weiss.

Generators for the Mod-p Cohomology of the Steinberg Summand of Thom Spectra Over $\mathrm {B}(\mathbb {Z}/p)^{n}$-Odd Primary Cases

Generators for the Mod-p Cohomology of the Steinberg Summand of Thom Spectra Over $\mathrm {B}(\mathbb {Z}/p)^{n}$-Odd Primary Cases

Algebraic elliptic cohomology and flops II: SL-cobordism

In this paper, we study the algebraic Thom spectrum MSL in Voevodsky's motivic stable homotopy category over an arbitrary perfect field k. Using the motivic Adams spectral sequence, we compute the geometric part of the η-completion of MSL. As an application, we study Krichever's elliptic genus with integral coefficients, restricted to MSL. We determine its image, and identify its kernel as the ideal generated by differences of SL-flops. This was proved by B. Totaro in the complex analytic setting. In the appendix, we prove some convergence properties of the motivic Adams spectral sequence.

Eilenberg–Mac Lane spectra as equivariant Thom spectra

We prove that the $G$-equivariant mod $p$ Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, $p$-power cyclic group $G$, generalizing a result of Behrens and the second author in the case of the group $C_2$. We also establish a construction of $\mathrm{H}\underline{\mathbb{Z}}_{(p)}$, and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant $\mathrm{H}\mathbb{F}_p$ arises as the Thom spectrum of a more than double loop map.

Exotic multiplications on periodic complex bordism

Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of $BU$. This presents an $\mathbb{E}_\infty$ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the $\mathbb{E}_\infty$-ring $MUP$. Here, we prove that these two $\mathbb{E}_\infty$-rings are in fact different, though the underlying $\mathbb{E}_2$-rings are equivalent. Nonetheless, we prove that both rings $\mathbb{E}_\infty$-orient $KU_2^{\wedge}$ and other forms of $K$-theory.

$\operatorname {H}(\mathbb {Z}/p^k)$ as a Thom spectrum and topological Hochschild homology

$\operatorname {H}(\mathbb {Z}/p^k)$ as a Thom spectrum and topological Hochschild homology

MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN RETRACTIVE SPACES

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set-level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding$\infty$-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted$K$-theory.

Modules over algebraic cobordism

AbstractWe prove that the$\infty $-category of$\mathrm{MGL} $-modules over any scheme is equivalent to the$\infty $-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite$\mathbf{P} ^1$-loop spaces, we deduce that very effective$\mathrm{MGL} $-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers.Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that$\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $is the$\mathbf{A} ^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for$n>0$,$\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $is the$\mathbf{A} ^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension$-n$.

THE UNIT MAP OF THE ALGEBRAIC SPECIAL LINEAR COBORDISM SPECTRUM

AbstractIn the joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^{1}$-loop spaces of motivic Thom spectra using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_{m}$-homotopy groups of motivic Thom spectra in terms of geometric generators and relations. Using this explicit description, we show that the unit map of the algebraic special linear cobordism spectrum induces an isomorphism on $\mathbb{G}_{m}$-homotopy sheaves.

E2 structures and derived Koszul duality in string topology

We construct an equivalence of $E_{2}$ algebras between two models for the Thom spectrum of the free loop space that are related by derived Koszul duality. To do this, we describe the functoriality and invariance properties of topological Hochschild cohomology.

Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable infinity category parametrized over objects of an infinity topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures. Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.

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

We show that the homotopy groups of a connective \(\mathbb {E}_k\)-ring spectrum with an \(\mathbb {E}_k\)-cell attached along a class \(\alpha \) in degree n are isomorphic to the homotopy groups of the cofiber of the self-map associated to \(\alpha \) through degree 2n. Using this, we prove that the \(2n-1\)st homotopy groups of Ravenel’s X(n) spectra are cyclic for all n. This further implies that, after localizing at a prime, \(X(n+1)\) is homotopically unique as the \(\mathbb {E}_1-X(n)\)-algebra with homotopy groups in degree \(2n-1\) killed by an \(\mathbb {E}_1\)-cell. Lastly, we prove analogous theorems for a sequence of \(\mathbb {E}_k\)-ring Thom spectra, for each odd k, which are formally similar to Ravenel’s X(n) spectra and whose colimit is also MU.