Stable Homotopy Research Articles

On a family involving R.L. Cohenʼs ζ-element

In 1981, R.L. Cohen constructed an infinite family of homotopy elements ζk∈π⁎(S) represented by h0bk∈ExtA3,2(p−1)(pk+1+1)(Z/p,Z/p) in the Adams spectral sequence, where p>2 and k⩾1. In this paper, we show that the composite map ζn−1β1βs+2 is nontrivial in the stable homotopy of spheres π2(p−1)[pn+(s+3)p+(s+2)]−7(S), where p⩾5, n>3 and 0⩽s<p−4.

The chromatic filtration of the Burnside category

AbstractThe Segal map connects the Burnside category of finite groups to the stable homotopy category of their classifying spaces. The chromatic filtrations on the latter can be used to define filtrations on the former. We prove a related conjecture of Ravenel's in some cases, and present counterexamples to the general statement.

The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime

AbstractThe chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the E2-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The E1-term of the spectral sequence is an Ext group of BP*BP-comodules. There is a sequence of Ext groups for nonnegative integers n with and there are Bockstein spectral sequences computing a module (n – s) from So far, a small number of the E1-terms are determined. Here, we determine the for p &gt; 2 and n &gt; 3 by computing the Bockstein spectral sequence with E1-term for s = 1, 2. As an application, we study the nontriviality of the action of α1 and β1 in the homotopy groups of the second Smith-Toda spectrum V(2).

Diagram spaces and symmetric spectra

We present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra. The main motivating examples are the I-spaces, which are diagrams indexed by finite sets and injections, and J-spaces, which are diagrams indexed by Quillen’s localization construction Σ−1Σ on the category Σ of finite sets and bijections.We show that the category of I-spaces provides a convenient model for the homotopy category of spaces in which every E∞ space can be rectified to a strictly commutative monoid. Similarly, the commutative monoids in the category of J-spaces model graded E∞ spaces.Using the theory of J-spaces we introduce the graded units of a symmetric ring spectrum. The graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures.

Motivic slices and coloured operads

Colored operads were introduced in the 1970's for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extend the slice filtration preserves highly structured objects such as algebras and modules. We use colored operads to give a precise solution to this problem. Our approach makes use of axiomatic setups which specialize to classical and motivic stable homotopy theory. Accessible t-structures are central to the development of the general theory. Concise introductions to colored operads and Bousfield (co)localizations are given in separate appendices.

Cellularization of structures in stable homotopy categories

AbstractWe describe the formal framework for cellularization functors in triangulated categories and study the preservation of ring and module structures under these functors in stable homotopy categories in the sense of Hovey, Palmieri and Strickland, such as the homotopy category of spectra or the derived category of a commutative ring. We prove that cellularization functors preserve modules over connective rings but they do not preserve rings in general (even if the ring is connective or the cellularization functor is triangulated). As an application of these results, we describe the cellularizations of Eilenberg–Mac Lane spectra and compute all acyclizations in the sense of Bousfield of the integral Eilenberg–Mac Lane spectrum.

The smash product for derived categories in stable homotopy theory

An E1 (or A∞) ring spectrum R has a derived category of modules DR. An E2 structure on R endows DR with a monoidal product ∧R. An E3 structure on R endows ∧R with a braiding. If the E3 structure extends to an E4 structure then the braided monoidal product ∧R is symmetric monoidal.

Products of Greek letter elements dug up from the third Morava stabilizer algebra

In [3], Oka and the second author considered the cohomology of the second Morava stabilizer algebra to study nontriviality of the products of beta elements of the stable homotopy groups of spheres. In this paper, we use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres: 1 t , ˇ2 t , h 1; 1; ˇ p p=pi tˇ1 and hˇ1;p; t i for t with p−t.t 1/ for a prime number p > 5 .

Modular representations and the homotopy of low rank p-local CW-complexes

Fix an odd prime p and let X be the p-localization of a finite suspended CW-complex. Given certain conditions on the reduced mod-p homology $${\widetilde H_*(X; \mathbb{Z}_p)}$$ of X, we use a decomposition of ΩΣX due to the second author and computations in modular representation theory to show there are arbitrarily large integers i such that ΩΣ i X is a homotopy retract of ΩΣX. This implies the stable homotopy groups of ΣX are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for ΣX. Under additional assumptions on $${\widetilde H_*(X; \mathbb{Z}_p)}$$ , we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of ΩΣX that has infinitely many finite H-spaces as factors.

Topological arbiters

This paper initiates the study of topological arbiters, a concept rooted in Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on $RP^2$, which reports the location of the essential 1-cycle. In contrast, we show that there exists an uncountable collection of topological arbiters in dimension 4. Families of arbiters, not induced by homology, are also shown to exist in higher dimensions. The technical ingredients underlying the four dimensional results are secondary obstructions to generalized link-slicing problems. For classical links in the 3-sphere the construction relies on the existence of nilpotent embedding obstructions in dimension 4, reflected in particular by the Milnor group. In higher dimensions novel arbiters are produced using nontrivial squares in stable homotopy theory. The concept of "topological arbiter" derives from percolation and from 4-dimensional surgery. It is not the purpose of this paper to advance either of these subjects, but rather to study the concept for its own sake. However in appendices we give both an application to percolation, and the current understanding of the relationship between arbiters and surgery. An appendix also introduces a more general notion of a multi-arbiter. Properties and applications are discussed, including a construction of non-homological multi-arbiters.

Fixed-orbit indices, I

Let G be a compact Lie group. Using methods from equivariant fibrewise stable homotopy theory, we study indices which measure algebraically (1) for a G-self-map f of a compact G-ENR, the set of G-orbits which are preserved by f and (2) for a G-vector field v on a closed smooth G-manifold, the set of points where v is parallel to the G-orbit through the point.

A note on numerators of Bernoulli numbers

The object of this short note is to give some observations on Bernoulli numbers and their function field analogs and point out ‘known’ counter-examples to a conjecture of Chowla. Bernoulli numbers Bn defined (for integer n > 1) by z/(e z − 1) = ∑ Bnz /n!, and their important cousins Bn/n, play interesting roles in many areas of mathematics. (Below we only restrict to these for n even, precisely the case when they are non-zero.) We mention some key words by which the reader can search: Power sums, Zeta special values, Eisenstein series, measures, p-adic L-functions, finite differences, combinatorics, Euler-Maclaurin formula, Todd classes in topology, Grothendieck-Hirzebruch-Riemann-Roch formula, K-theory of integers, Stable homotopy, Bhargava factorial associated to the set of primes, Kummer-HerbrandRibet theorems in cyclotomic theory, Kervaire-Milnor formula for diffeomorphism classes of exotic spheres. Their factorization is of interest, the denominators (which show up explicitly in the third-fourth items from the end) are well understood via theorems of von-Staudt, but the numerators (which show up explicitly in the last two items above) are mysterious and connected to many interesting phenomena. In one of the rare lapses, Ramanujan, in his very first paper [R1911, (14), (18) and Sec. 12], claimed to have proved (editors downgrade it to a conjecture) that the numerator Nn of Bn/n is always a prime, when it was already known since Kummer (in Fermat’s last theorem connection) that ‘irregular’ prime 37 is a proper divisor of N32, and even N20 is composite. In [C1930], Chowla showed that Ramanujan’s claim had infinity of counter-examples. Note that this also follows from one counterexample and the Kummer congruences (recalled below) for that prime! Interestingly, in his last paper [CC1986], Chowla (jointly with his daughter) asks as unsolved problem whether the numerator is always square-free. (This is also mentioned in the nice survey article by Murtys and Williams on Chowla’s work in Vol. 1 of [C1999], where the author learned about it.) Theorem 1. Chowla’s conjecture stated above has infinity of counter-examples. In fact, for any given irregular prime p less than 163 million, and given arbitrarily large k, there is n such that p divides Nn. Proof. Using the tables (or the reader can try to check directly!) giving factorizations of Bn/n, for example the table by Wagstaff at the Bernoulli web page www.bernoulli.org, we see that 37 divides N284. Now recall the well-known Kummer congruences that the value of (1 − p)Bn/n modulo p depends only on (even) n modulo pk−1(p − 1), for n not divisible by p − 1. The first claim follows by taking p = 37. Supported in part by NSA grant H98230-10-1-0200.

On genera of polyhedra

Abstract We consider the stable homotopy category S of polyhedra (finite cell complexes). We say that two polyhedra X,Y are in the same genus and write X ∼ Y if X p ≅ Y p for all prime p, where X p denotes the image of Xin the localized category S p. We prove that it is equivalent to the stable isomorphism X∨B 0 ≅Y∨B 0, where B 0 is the wedge of all spheres S n such that π nS(X) is infinite. We also prove that a stable isomorphism X ∨ X ≅ Y ∨ X implies a stable isomorphism X ≅ Y.

Sheaves and K-theory for [formula omitted]-schemes

This paper is devoted to the open problem in F1-geometry of developing K-theory for F1-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for M0-schemes and F1-schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over F1. Special attention is paid to two aspects particular to F1-geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillenʼs Q-construction to a definition of G-theory and K-theory for F1-schemes. A comparison with Waldhausenʼs S•-construction yields the ring structure of K-theory. In particular, we generalize Deitmarʼs K-theory of monoids and show that K⁎(SpecF1) realizes the stable homotopy of the spheres as a ring spectrum.

Bar-cobar duality for operads in stable homotopy theory

We extend bar-cobar duality, defined for operads of chain complexes by Getzler and Jones, to operads of spectra in the sense of stable homotopy theory. Our main result is the existence of a Quillen equivalence between the category of reduced operads of spectra (with the projective model structure) and a new model for the homotopy theory of cooperads of spectra. The crucial construction is of a weak equivalence of operads between the Boardman-Vogt W-construction for an operad P, and the cobar-bar construction on P. This weak equivalence generalizes a theorem of Berger and Moerdijk that says the W- and cobar-bar constructions are isomorphic for operads of chain complexes. Our model for the homotopy theory of cooperads is based on `pre-cooperads'. These can be viewed as cooperads in which the structure maps are zigzags of maps of spectra that satisfy coherence conditions. Our model structure on pre-cooperads is such that every object is weakly equivalent to an actual cooperad, and weak equivalences between cooperads are detected in the underlying symmetric sequences. We also interpret our results in terms of a `derived Koszul dual' for operads of spectra, which is analogous to the Ginzburg-Kapranov dg-dual. We show that the `double derived Koszul dual' of an operad P is equivalent to P (under some finiteness hypotheses) and that the derived Koszul construction preserves homotopy colimits, finite homotopy limits and derived mapping spaces for operads.

Uniqueness of Massey Products on the Stable Homotopy of Spheres

The product on the stable homotopy ring of spheres π*scan be defined by composing, smashing or joining maps. Each of these three points of view is used in Section 2 to define Massey products on π*s. In fact we define composition and smash Massey products (x1, … , xt)where X1, … ,xt-1∈ π*s, xt∈ π*(E) and E is a spectrum. In Theorem 3.2, we prove that these three types of Massey products are equal. Consequently, a theorem which is easy to prove for one of these Massey products is also valid for the other two. For example, [3, Theorem 8.1] which relates algebraic Massey products in the Adams spectral sequence to Massey smash products in π*s is now also valid for Massey composition products in π*s

Multiplicative maps from HZ to a ring spectrum R–-a naive version

The paper is devoted to study the space of multiplicative maps from the Eilenberg-MacLane spectrum $H\Z$ to an arbitrary ring spectrum $R$. We try to generalize the approach of Schwede from Formal groups and stable homotopy of commutative rings, where the special case of the mentioned above problem was solved in full generality. Among other results we propose a definition of a formal group law in any ring spectrum, which might be of interest in its own.

Bott periodicity for inclusions of symmetric spaces

When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unitary group. Standard inclusions between other classical Riemannian symmetric spaces are considered as well. An application to homotopy theory is also discussed.

Stable frames in model categories

We develop a stable analogue to the theory of cosimplicial frames in model categories; this is used to enrich all homotopy categories of stable model categories over the usual stable homotopy category and to give a different description of the smash product of spectra which is compared with the known descriptions; in particular, the original smash product of Boardman is identified with the newer smash products coming from a symmetric monoidal model of the stable homotopy category.

Monoidality of Frankeʼs exotic model

We discuss the monoidal structure on Frankeʼs algebraic model for the K ( p ) -local stable homotopy category at odd primes and show that its Picard group is isomorphic to the integers.