Milnor Conjecture Research Articles

On generalized Milnor's conjecture for abelian topological groups

We consider an extension of Milnor's conjecture [15] from Lie groups to arbitrary topological groups and prove its generalized version for locally compact finite dimensional abelian groups.We present counterexamples to extended Milnor's conjecture in the class of abelian topological groups whose underlying space is a CW-complex.

Brieskorn spheres, cyclic group actions and the Milnor conjecture

AbstractIn this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants defined by the first author satisfy for torus knots, whenever is a prime not dividing . Since is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere does not extend smoothly to any homology 4‐ball bounding . In the case of a non‐free cyclic group action of prime order, we prove that if the rank of is greater than times the rank of , then the ‐action on does not extend smoothly to any homology 4‐ball bounding . Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres.

On a refinement of the non-orientable 4-genus of Torus knots

In formulating a non-orientable analogue of the Milnor Conjecture on the 4 4 -genus of torus knots, Batson [Math. Res. Lett. 21 (2014), pp. 423–436] developed an elegant construction that produces a smooth non-orientable spanning surface in B 4 B^4 for a given torus knot in S 3 S^3 . While Lobb [Math. Res. Lett. 26 (2019), pp. 1789] showed that Batson’s surfaces do not always minimize the non-orientable 4 4 -genus, we prove that they do minimize among surfaces that share their normal Euler number. We also determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson’s surfaces are non-orientable 4 4 -genus minimizers.

Knot Floer homology and relative adjunction inequalities

We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These “relative adjunction inequalities” improve the adjunction inequalities for closed surfaces which have been instrumental in many topological applications of gauge theory. The relative inequalities refine the latter by incorporating numerical invariants of knots in the boundary associated to Heegaard Floer homology classes determined by the 4-manifold. As a corollary, we produce a host of concordance invariants for knots in a general 3-manifold, one such invariant for every non-zero Floer class. We apply our results to produce analogues of the Ozsváth–Szabó–Rasmussen concordance invariant for links, allowing us to reprove the link version of the Milnor conjecture, and, furthermore, to show that knot Floer homology detects strongly quasipositive fibered links.

Three real Artin-Tate motives

We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by the second-named author. So the novelty is concentrated at the prime 2, where modular representation theory enters the picture via work of Positselski, based on Voevodsky's resolution of the Milnor Conjecture. With coefficients in k=Z/2, our spectrum becomes homeomorphic to the spectrum of the derived category of filtered kC2-modules with a peculiar exact structure, for the cyclic group C2=Gal(C/R). This spectrum consists of six points organized in an interesting way. As an application, we find exactly fourteen classes of mod-2 real Artin-Tate motives, up to the tensor-triangular structure. Among those, three special motives stand out, from which we can construct all others. We also discuss the spectrum of Artin motives and of Tate motives.

On a nonorientable analogue of the Milnor conjecture

The nonorientable 4-genus $\gamma_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of $\gamma_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound for $\gamma_4$ formulated by Ozsv\'ath, Stipsicz, and Szab\'o. As a side product we obtain new closed formulas for the signature of torus knots.

Four-dimensional aspects of tight contact 3-manifolds

We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in [Formula: see text] An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces in [Formula: see text] with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth-Szabó contact invariant.

A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class

We give a description of Rasmussen’s [Formula: see text]-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram [Formula: see text], for any choice of an integral domain [Formula: see text] and a non-zero, non-invertible element [Formula: see text], we define the [Formula: see text]-divisibility [Formula: see text] of Lee’s canonical class of [Formula: see text], and prove that a combination of [Formula: see text] and some elementary properties of [Formula: see text] yields a link invariant [Formula: see text]. Each [Formula: see text] possesses properties similar to [Formula: see text], which in particular reproves the Milnor conjecture. If we restrict to knots and take [Formula: see text], then our invariant coincides with [Formula: see text].

Nonnegative Ricci curvature, stability at infinity and finite generation of fundamental groups

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $\pi_1(M)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and the unique tangent cone at infinity.

On Farrell–Tate cohomology of SL2 over S-integers

In this paper, we provide number-theoretic formulas for Farrell–Tate cohomology for SL2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups.We expose three applications, to (I) detection questions for the Quillen conjecture,(II) the existence of transfers for the Friedlander–Milnor conjecture,(III) cohomology of SL2 over number fields.

Foliations of codimension one and Milnor's conjecture

Foliations of codimension one and Milnor's conjecture

A proof of Milnor conjecture in dimension 3

Abstract We present a proof of the Milnor conjecture in dimension 3 based on Cheeger–Colding theory on limit spaces of manifolds with Ricci curvature bounded below. It is different from [17] that relies on minimal surface theory.

An Overview of Motivic Homotopy Theory

Motivic homotopy theory was constructed by Morel and Voevodsky in the 1990s. It led to such striking applications as the solution of the Milnor conjecture and the Bloch-Kato conjecture on the Galois symbol. Since then the theory has turned to a systematic study of its basic invariants and structures, as well as providing numerous new applications. Morel, Cisinski-Deglise, and Rondigs-Ostvaer have described the relation of the motivic stable homotopy category to Voevodskys triangulated category of motives. Morel’s computation of the endomorphism of the sphere spectrum points out a close relationship to quadratic forms, and has given rise to the construction of interesting new oriented cycle theories. Asok and Fasel have applied computations of unstable motivic homotopy groups to stability problems for algebraic vector bundles. Isaksen and others have computed motivic versions of Adams-Novikov and Adams spectral sequences, and used this information to improve known computations of these spectral sequences in classical homotopy theory. We will discuss the basic ideas going into the construction of motivic homotopy theory and some of these results and applications.

Slices of hermitian K–theory and Milnor’s conjecture on quadratic forms

We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.

The Rasmussen invariant and the Milnor conjecture

These notes were written for a series of lectures on the Rasmussen invariant and the Milnor conjecture, given at Winter Braids IV in February 2014.

The Witt ring of a curve with good reduction over a non-dyadic local field

In this work, we present a generalization to varieties and sheaves of the fundamental ideal of the Witt ring of a field by defining a sheaf of fundamental ideals I˜ and a sheaf of Witt rings W˜ in the obvious way. The Milnor conjecture then relates the associated graded of W˜ to Milnor K-theory and so allows the classical invariants of a bilinear space over a field to be extended to our setting using étale cohomology. As an application of these results, we calculate the Witt ring of a smooth curve with good reduction over a non-dyadic local field.

Milnor K-theory and the graded representation ring

AbstractLet F be a field, let G = Gal(/F) be its absolute Galois group, and let R(G,k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k*(F) to the graded ring grR(G,k) associated to Grothendieck's γ-filtration. We study this map in particular cases, as well as a related map involving the W-group of F, rather than G. The latter is an isomorphism in all cases considered.Naturally this echoes the Milnor conjecture (now a theorem), which states that k*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring grW(F).The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the corresponding K-theory ring KO(X) of the topological space X, again with respect to the γ-filtration, and whose target is a certain subquotient of H*(X, F2).In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.

Rationality problems and conjectures of Milnor and Bloch–Kato

AbstractWe show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

THE STRONG CHOWLA–MILNOR SPACES AND A CONJECTURE OF GUN, MURTY AND RATH

In a recent work, Gun, Murty and Rath formulated the Strong Chowla–Milnor conjecture and defined the Strong Chowla–Milnor space. In this paper, we proved a non-trivial lower bound for the dimension of these spaces. We also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of this lower bound will lead to irrationality of both ζ(k) and ζ(k)/πk for all odd positive integers k. Following Gun, Murty and Rath, we define generalized Zagier spaces Vp(K) for multiple zeta values over a number field K. We prove that the dimension of V4d+2(K) for d ≥ 1, is at least 2, assuming a conjecture of Gun, Murty and Rath.

Grid diagrams and the Ozsváth-Szabó tau-invariant

We use grid diagrams to investigate the Ozsvath-Szabo concordance invariant tau, and to prove that |tau(K_1)-tau(K_2)|<=g, whenever there is a genus g knot cobordism joining K_1 to K_2. This leads to an entirely grid diagram-based proof of Kronheimer-Mrowka's theorem, formerly known as the Milnor conjecture.