Thom Spectra Research Articles

A filtration of unoriented cobordism

We give a filtration of BO that recovers the polynomial generators of H⁎BO one at a time, and that the corresponding Thom spectra filter the unoriented cobordism ring π⁎(MO) and recovers the polynomial generators of this ring one at a time.

Fold maps, framed immersions and smooth structures

For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fold maps splits off (q-1)/2 cobordism groups of framed immersions. The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds.

Cobordism categories of manifolds with corners

In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if Cob d , ⟨ k ⟩ \text {Cob}_{d,\langle k\rangle } is the category whose objects are a fixed dimension d d , with corners of codimension ≤ k \leq \;k , then we identify the homotopy type of the classifying space B Cob d , ⟨ k ⟩ B\text {Cob}_{d,\langle k\rangle } as the zero space of a homotopy colimit of a certain diagram of the Thom spectra. We also identify the homotopy type of the corresponding cobordism category when an extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss (2009), and their proofs are an adaptation of the methods of their paper. As an application we describe the homotopy type of the category of open and closed strings with a background space X X , as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez (2006) and Hanbury.

Realizing a complex of unstable modules

In a preceding article~[7] the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra.

Higher topological Hochschild homology of Thom spectra

In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological Andre-Quillen homology.

Topological Hochschild homology of Thom spectra and the free loop space

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ.

Monoids of moduli spaces of manifolds

We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_\theta$ of closed smooth (d-1)-manifolds and smooth d-dimensional cobordisms, equipped with generalised orientations specified by a fibration $\theta : X \to BO(d)$. The main result of GMTW is a determination of the homotopy type of the classifying space $BC_\theta$. The goal of the present paper is a systematic investigation of subcategories $D$ of $C_\theta$ having classifying space homotopy equivalent to that of $C_\theta$, the smaller such $D$ the better. We prove that in most cases of interest, $D$ can be chosen to be a homotopy commutative monoid. As a consequence we prove that the stable cohomology of many moduli spaces of surfaces with $\theta$-structure is the cohomology of the infinite loop space of a certain Thom spectrum. This was known for certain special $\theta$, using homological stability results; our work is independent of such results and covers many more cases.

Topological Hochschild homology of Thom spectra which are E ∞ -ring spectra

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E ∞ -ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits and is in fact a left adjoint. We prove a splitting result THH(M f) ≃ eq Mf ∧ BX + , which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X → BG is only a homotopy commutative A ∞ map, provided that the induced multiplication on Mf extends to an E ∞ -ring structure; this permits us to recover Bokstedt's calculation of THH(Hℤ).

Characteristic cohomotopy classes for families of 4-manifolds

Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer-Furuta invariants. The definition is given in the context of parametrised stable homotopy theory, but an interpretation in terms of characteristic cohomotopy classes on Thom spectra associated to the classifying spaces of complex spin diffeomorphism groups is given as well. The theory is illustrated with families of K3 surfaces and mapping tori of diffeomorphisms. It is also related to equivariant invariants.

Fibration of classifying spaces in the cobordism theory of singular maps

In the cobordism theory of singular smooth maps there exist classifying spaces (analogues of Thom spectra) depending on the set of allowed singularity types. The so-called “key fibration” introduced by A. Szűcs connects these classifying spaces for different sets of allowed singularities. Here we prove the existence of such a fibration using a new, more simple and general argument than that of Szűcs. This makes it possible to extend the range of applications to some negative codimension maps.

Topological Hochschild homology of twisted group algebras

We show that the topological Hochschild homology spectrum of a twisted group algebra THH ⁡ ( A τ [ G ] ) \operatorname {THH}(A^{\tau }[G]) is the Thom spectrum associated with a parametrized orthogonal spectrum E ( A , G ) E(A,G) . We then analyze the structure of the parametrized orthogonal spectrum E ( A , G ) E(A,G) and show that it is locally trivial.

Résolution de certains modules instables et fonction de partition de Minc

One constructs minimal injective resolutions for certain unstable modules that appear to be the mod 2 cohomology of Thom spectra. The terms of the resolution are tensor products of Brown–Gitler modules and Steinberg modules introduced by S. Mitchell and S. Priddy. A combinatorial result of Andrews shows that the alternating sum of the Poincaré series of the considered modules is zero. One gives homotopical applications of this result. To cite this article: D.H.H. Nguyen et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Thom spectra that are symmetric spectra

We analyze the functorial and multiplicative properties of the Thom spectrum functor in the setting of symmetric spectra and we establish the relevant homotopy invariance.

The homotopy ofMStringandMU⟨6⟩ at large primes

Recall that a String manifold is a Spin manifold M together with a trivialization of the class usually denoted p1(M)/2; this is a characteristic class, defined only for Spin manifolds, so that twice it is the usual first Pontrjagin class. The bordism spectrum of String manifolds is called MString or MO〈8〉; it is the Thom spectrum of the map BO〈8〉 −→ BO of the 7-connected cover of BO to BO. Similarly, MU〈6〉 is the bordism spectrum associated with the 5-connected cover BU〈6〉 −→ BU of BU .

Quasisymmetric functions from a topological point of view

It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions $\mathsf {Symm}$. We offer the cohomology of the space $\Omega \Sigma {\mathsf C} P^{\infty}$ as a topological model for the ring of quasisymmetric functions $\mathsf {QSymm}$. We exploit standard results from topology to shed light on some of the algebraic properties of $\mathsf {QSymm}$. In particular, we reprove the Ditters conjecture. We investigate a product on $\Omega \Sigma {\mathsf C} P^{\infty}$ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of $BU$. The canonical Thom spectrum over $\Omega \Sigma {\mathsf C} P^{\infty}$ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.

Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum build from the canonical bundle over the Grassmannians of 2-planes in R^{n+2}. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i--this is the non-oriented analogue of the Mumford conjecture.

Generalized string topology operations

We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally the Thom spectrum E^{-TM} is a ring spectrum. Similarly a fiberwise module over E defines a module over E^{-TM}. Fiberwise monoids include adjoint bundles of principal bundles, and the construction is natural with respect to maps of principal bundles. This naturality implies homotopy invariance of the algebra structure on H_*(LM) arising from the loop product. If M=BG is the infinite dimensional classifying space of a compact Lie group, then we get a well-defined pro-ring spectrum, which we define to be the string topology of BG. If E has a fiberwise action of the little n-cubes operad then E^{-TM} is an E_n-ring spectrum. This gives homology operations combining Dyer-Lashof operations on the fiber and Steenrod operations on the base. We give several examples where the new operations give homological insight, borrowed from knot theory, complex geometry, gauge theory, and homotopy theory.

A duality between string topology and the fusion product in equivariant K-theory

Let $G$ be a compact Lie group. Let $G \to E \to M$ be a principal $G$-bundle over a closed manifold $M$, and $G \to Ad(E) \to M$ its adjoint bundle. In this paper we describe a new Frobenius algebra structure on $h_*(Ad(E)),$ where $h_*$ is an appropriate generalized homology theory. Recall that a Frobenius algebra has both a product and a coproduct. The product in this new Frobenius algebra is induced by the string topology product. In particular, the product can be defined when $G$ is any topological group and in the case that $E$ is contractible it is precisely the Chas-Sullivan string product on $H_*(LM).$ We will show that the coproduct is induced by completing the untwisted Freed-Hopkins-Teleman fusion product. Indeed, when $M$ is replaced by $BG$ and $h_*$ is $K$-theory the coproduct is the completion of the Freed-Hopkins-Teleman fusion structure. We will then show that this duality between the string and fusion products is realized by a Spanier-Whitehead duality between certain Thom spectra of virtual bundles over $Ad(E)$.

Arbeitsgemeinschaft mit aktuellem Thema: Algebraic Cobordism

Over the years, many different types and flavors of cohomology theories for algebraic varieties have been constructed. Theories like étale cohomology or de Rham cohomology provide algebraic versions of the topological theory of singular cohomology. The Chow ring and algebraic K_0 are other (partial) examples, more directly tied to algebraic geometry. The partial theory K_0^{alg} was extended to a full theory with the advent of Quillen's higher algebraic K -theory. It took considerably longer for the Chow ring to be extended to motivic cohomology. In the process of doing so, Voevodsky developed his category of motives, and this construction was put in a more general setting with the development by Morel–Voevodskyof of \mathbb{A}^1 homotopy theory. This enabled a systematic construction of cohomology theories on algebraic varieties, with algebraic K -theory and motivic cohomology being only two fundamental examples. These two cohomology theories have in common the existence of a good theory of push-forward maps for projective morphisms. Not all cohomology theories have this structure, those that do are called oriented . In the Morel–Voevodsky stable homotopy category, the universal oriented theory is represented by the \mathbb{P}^1 -spectrum M\mathbb G\ell , an algebraic version of the classical Thom spectrum MU . The corresponding cohomology theory M\mathbb G\ell^{*,*} is called higher algebraic cobordism . In an attempt to better understand the theory M\mathbb G\ell^{*,*} , Levine and Morel constructed a theory of algebraic cobordism \Omega^* . This is (conjecturally) related to M\mathbb G\ell^{*,*} as the classical Chow ring \mathrm{CH}^* is to motivic cohomology and like \mathrm{CH}^* , \Omega^* has a purely algebro-geometric description. In addition to giving some insight into M\mathbb G\ell^{*,*} , \Omega^* gives a simultaneous presentation of both \mathrm{CH}^* and K_0 , exhibiting K_0 as a deformation of \mathrm{CH}^* . \Omega^* has also been used to give conceptually simple proofs of various “degree formulas” first formulated by Rost. These degree formulas have been used in the study of Pfister quadrics and norm varieties, properties of which are used in the proofs of the Milnor conjecture and the Bloch–Kato conjecture. In this workshop, we describe aspects of the topological theory of complex cobordism which are important for algebraic cobordism (Lectures 1-3) and give the construction of \Omega^* and proofs of its fundamental properties (Lectures 4-7). In lectures 8-11, we show how K_0 and \mathrm{CH}^* are described by \Omega^* , how \Omega^* recovers the universal formal group law, give the proof the generalized degree formula for \Omega^* and use this to proof the degree formula for the Segre class. Additional applications to Steenrod operations, further degree formulas and the use of these in the study of quadrics and other varietes is given in lectures 12 and 13. Lectures 14 and 15 concern the construction of funtorial pull-backs in algebraic cobordism. The two concluding lectures (16 and 17) give a quick sketch of the Morel–Voevodsky \mathbb{A}^1 stable homotopy category and describe what we know about M\mathbb G\ell and its relation to motivic cohomology and \Omega^* . The workshop Algebraic Cobordism ,organised by Marc Levin (Boston) and Fabien Morel (München) was held April 4th–April 8th, 2005. This meeting was well attended with 55 participants.

A lower bound for coherences on the Brown–Peterson spectrum

We provide a lower bound for the coherence of the homotopy commutativity of the Brown-Peterson spectrum, BP, at a given prime p and prove that it is at least (2p^2 + 2p - 2)-homotopy commutative. We give a proof based on Dyer-Lashof operations that BP cannot be a Thom spectrum associated to n-fold loop maps to BSF for n=4 at 2 and n=2p+4 at odd primes. Other examples where we obtain estimates for coherence are the Johnson-Wilson spectra, localized away from the maximal ideal and unlocalized. We close with a negative result on Morava-K-theory.