Stable Homotopy Research Articles

Minimum numbers and Wecken theorems in topological coincidence theory. I

Minimum numbers measure the obstruction to removing coincidences of two given maps (between smooth manifolds M and N of dimensions m and n, resp.). In this paper, we compare them to four distinct types of Nielsen numbers. These agree with the classical Nielsen number when m = n (e.g., in the fixed point setting where M = N and one of the maps is the identity map). However, in higher codimensions, m − n > 0, their definitions and computations involve distinct aspects of differential topology and homotopy theory. We develop tools which help us (1) to decide when a minimum number is equal to a Nielsen number (“Wecken theorem”), and (2) to determine Nielsen numbers. Here certain homotopy theoretical criteria play a central role. E.g., failures of the “Wecken condition” (cf. Definition 1.18 below) can have very interesting geometric consequences. The selfcoincidence case where the two maps are homotopic turns out to be particularly illuminating. We give many concrete applications in special settings where M or N are spheres, spherical space forms, projective spaces, tori, Stiefel manifolds or Grassmannians. Already in the simplest examples an important role is played, e.g., by Kervaire invariants, all versions of Hopf invariants (à la James, Hilton, Ganea, . . . ) and the elements in the stable homotopy of spheres defined by invariantly framed Lie groups.

On the chromatic $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for an odd prime

Ravenel introduced spectra $T(m)$ for $m\ge 0$ interpolating the Brown-Peterson spectrum $BP$ and the sphere spectrum $S$ in Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, Providence, 2004. Since the homotopy groups of $BP$ are well known, it is interesting to study differences among the homotopy groups of $T(m)$'s to study the homotopy groups of spheres. He also introduced the localization functor $L_n$ on the stable homotopy category in "Localization with respect to certain periodic homology theories," Amer. J. Math. 106 (1984), 351–414. To study the difference of $L_nT(m)$'s for a fixed integer $n$, we consider the corresponding chromatic $E_1$-term $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for each $m$, and determine it for $m+1\ge (n-2)(n-1)$ in this paper. The results show that the structures depend on a integer $\Lt[\dfrac{m+1}{n-1}\Rt]$. Here $[x]$ denotes the greatest integer that does not exceed $x$.

On R L Cohen’sζ–element

Let p be a prime greater than three. In the p–local stable homotopy groups of spheres, R L Cohen constructed the infinite ζ–element ζn−1∈π2pn+1−2pn+2p−5(S) of order p. In the stable homotopy group π2pn+1−2pn+2p2−3(V(1)) of the Smith–Toda spectrum V(1), X Liu constructed an essential element ϖk for k≥3. Let βs∗=j0j1βs∈[V(1),S]2sp2−2s−2p denote the Spanier–Whitehead dual of the generator βs′′=βsi1i0∈π2sp2−2s(V(1)), which defines the β–element βs. Let ξs,k=βs−1∗ϖk. In this paper, we show that the composite of R L Cohen’s ζ–element ζn−1 with ξs,n is nontrivial, where n>4 and 1<s<p−1. As a corollary, ξs,n is also nontrivial for 1<s<p−1.

Symmetric Joins and Weighted Barycenters

Abstract Given a space X, we study the homotopy type of Bn(X) the space obtained as the “union of all (n−1)-simplexes spanned by points in X” or the space of “formal barycenters of weight n or less” of X. This is a space encountered in non-linear analysis under the name of space of barycenters or in differential geometry in the case n = 2 as the space of chords. We first relate this space to a more familiar symmetric join construction and then determine its stable homotopy type in terms of the symmetric products on the suspension of X. This leads to a complete understanding of the homology of Bn(X) as a functor of X, and to an expression for its Euler characteristic given in terms of X. A sharp connectivity theorem is also established. Finally the case of spheres S is studied in details and the homotopy type of Bn(S ) is described, generalizing in this way an early and beautiful result of James, Thomas, Toda and Whitehead.

Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence

AbstractWe describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulæ for the computation of the E3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

Erratum: Corrections to the paper “geometric approach to stable homotopy groups of spheres. The adams–hopf invariants”

Erratum: Corrections to the paper “geometric approach to stable homotopy groups of spheres. The adams–hopf invariants”

The Slice Filtration and Grothendieck-Witt Groups

Let k be a perfect field of characteristic different from two. We show that the filtration on the Grothendieck-Witt group GW(k) induced by the slice filtration for the sphere spectrum in the motivic stable homotopy category is the I-adic filtration, where I is the augmentation ideal in GW(k).

Derived Mackey Functors

For a finite group $G$, the so-called $G$-Mackey form an abelian category $M(G)$ that has many applications in the study of $G$-equivariant stable homotopy. One would expect that the category $D(M(G))$ would be similarly important as the homological counterpart of the $G$-equivariant stable homotopy category. It turns out that this is not so -- $D(M(G))$ is pathological in many respects. We propose and study a replacement for $D(M(G))$, a certain triangulated category $DM(G)$ of derived Mackey functors that contains $M(G)$ but is different from $D(M(G))$. We show that standard features of the $G$-equivariant stable homotopy category such as the fixed points of two types have exact analogs for the category $DM(G)$.

On the orientability of the slice filtration

Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\mathcal{SH}$ are strict modules over Voevodsky’s algebraic cobordism spectrum. We also show that the zero slice of any commutative ring spectrum in $\mathcal{SH}$ is an oriented ring spectrum in the sense of Morel, and that its associated formal group law is additive. As a consequence, we deduce that with rational coefficients the slices are in fact motives in the sense of Cisinski-Deglise and have transfers if the base scheme is excellent. This proves a conjecture of Voevodsky.

Symmetric homology of algebras

The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to stable homotopy theory via $HS_*(k[\Gamma]) \cong H_*(\Omega\Omega^{\infty} S^{\infty}(B\Gamma); k)$. Two chain complexes that compute $HS_*(A)$ are constructed, both making use of a symmetric monoidal category $\Delta S_+$ containing $\Delta S$. Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, $Sym^{(p)}_*$. $Sym^{(p)}$ is isomorphic to the suspension of the cycle-free chessboard complex $\Omega_{p+1}$ of Vre\'{c}ica and \v{Z}ivaljevi\'{c}, and so recent results on the connectivity of $\Omega_n$ imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the $k\Sigma_{p+1}$--module structure of $Sym^{(p)}$ are devloped. A partial resolution is found that allows computation of $HS_1(A)$ for finite-dimensional $A$ and some concrete computations are included.

Cohomological obstruction theory for Brauer classes and the period-index problem

AbstractLet U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(Uét,ℂm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,ℂm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then , where [] is the integer part of , whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

Weight structures vs.t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)

AbstractIn this paper we introduce a new notion ofweight structure (w)for a triangulated categoryC; this notion is an important natural counterpart of the notion oft-structure. It allows extending several results of the preceding paper [Bon09] to a large class of triangulated categories and functors.Theheartofwis an additive categoryHw⊂C. We prove that a weight structure yields Postnikov towers for anyX∈ObjC(whose 'factors'Xi∈ObjHw). For any (co)homological functorH:C→A(Ais abelian) such a tower yields aweight spectral sequenceT : H(Xi[j]) ⇒H(X[i + j]); Tis canonical and functorial inXstarting fromE2.Tspecializes to the usual (Deligne) weight spectral sequences for 'classical' realizations of Voevodsky's motivesDMeffgm(if we considerw = wChowwithHw=Choweff) and to Atiyah-Hirzebruch spectral sequences in topology.We prove that there often exists an exact conservative weight complex functorC→K(Hw). This is a generalization of the functort:DMeffgm→Kb(Choweff) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soulé). We prove thatK0(C) ≅K0(Hw) under certain restrictions.We also introduce the concept of adjacent structures: at-structure isadjacenttowif their negative parts coincide. This is the case for the Postnikovt-structure for the stable homotopy categorySH(of topological spectra) and a certain weight structure for it that corresponds to the cellular filtration. We also define a new (Chow)t-structuretChowforDMeff_⊃DMeffgmwhich is adjacent to the Chow weight structure. We haveHtChow≅ AddFun(Choweffop,Ab);tChowis related to unramified cohomology. Functors left adjoint to those that aret-exact with respect to somet-structures are weight-exact with respect to the corresponding adjacent weight structures, and vice versa. Adjacent structures identify two spectral sequences converging toC(X,Y): the one that comes from weight truncations ofXwith the one coming fromt-truncations ofY(forX,Y∈ObjC). Moreover, the philosophy of adjacent structures allows expressing torsion motivic cohomology of certain motives in terms of the étale cohomology of their 'submotives'. This is an extension of the calculation of E2of coniveau spectral sequences (by Bloch and Ogus).

The beta elementsβtp2∕rin the homotopy of spheres

In [1], Miller, Ravenel and Wilson defined generalized beta elements in the E2-term of the Adams-Novikov spectral sequence converging to the stable homotopy groups of spheres, and in [5], Oka showed that the beta elements of the form βtp2/r for positive integers t and r survives to the stable homotopy groups at a prime p > 3, when r ≤ 2p − 2 and r ≤ 2p if t > 1. In this paper, we expand the condition so that βtp2/r for r ≤ p2 − 3 survives to the stable homotopy groups.

Stable Projective Homotopy Theory of Modules, Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.

Gross-Hopkins duality and the Gorenstein condition

AbstractGross and Hopkins have proved that in chromatic stable homotopy, Spanier-Whitehead duality nearly coincides with Brown-Comenetz duality. We give a conceptual interpretation of this phenomenon in terms of a Gorenstein condition [8] for maps of ring spectra.

Matrix problems, triangulated categories and stable homotopy types

We show how the matrix problems can be used in studying triangulated categories. Then we apply the general technique to the classi cation of stable homotopy of polyhedra, and out the \representation types of such problems and give a description of stable homotopy in nite and tame cases.

Continuous group actions on profinite spaces

For a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point spaces and for stable homotopy groups of homotopy orbit spaces. Our main example is the Galois action on profinite étale topological types of varieties over a field. One motivation is to understand Grothendieck’s section conjecture in terms of homotopy fixed points.

Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra has a triangular spectrum much bigger than the Zariski spectrum of Z. We also give a first discussion of the spectrum in two new examples, namely equivariant KK ‐theory and stable A 1 ‐ homotopy theory. 18E30; 14F05, 19K35, 20C20, 55P42, 55U35

Orientable homotopy modules

We prove a conjecture of Morel identifying Voevodsky’s homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy $t$-structure and have a trivial action of the Hopf map. This is done by relating these two kind of objects to Rost’s cycle modules. Applications to algebraic cobordism and construction of cycle classes are given.

Motivic strict ring models for 𝐾-theory

It is shown that theKK-theory of every noetherian base scheme of finite Krull dimension is represented by a commutative strict ring object in the setting of motivic stable homotopy theory. The adjective ‘strict’ is used here in order to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. An analogous topological result follows by running the same type of arguments as in the motivic setting.