arXiv subjects

Kevin Hutchinson

Publications and source records attributed to Kevin Hutchinson.

17 recordsLinked to original sources

arXiv2026

Bloch Groups of Rings

We give a definition of (refined) Bloch groups of general commutative rings which agrees with the standard definition in the case of local rings whose residue field has at least $4$ elements. Under appropriate conditions on a ring $A$, satisfied by any field or local ring, these groups are closely related to third homology of $\mathrm{SL}_2(A)$ and to indecomposable $K_3$ of $A$. We analyze these conditions. We calculate the Bloch groups of $\mathbb{F}_2,\mathbb{F}_3,\mathbb{Z}$ and $\mathbb{Z}[\frac{1}{2}]$.

math.KT↗

arXiv2024

The Chern class for $K_3$ and the cyclic quantum dilogarithm

In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in their recent paper in Ann. Sci. École Normale Sup. that $R_ζ=c_ζ^2$, where $R_ζ$ is their map on $K_3$ defined using the cyclic quantum dilogarithm and $c_ζ$ is the Chern class map on $K_3$.

math.KT↗

arXiv2022

$\mathrm{GE}_2$-rings and a graph of unimodular rows

For a commutative ring $A$ we consider a related graph, $Γ(A)$, whose vertices are the unimodular rows of length $2$ up to multiplication by units. We prove that $Γ(A)$ is path-connected if and only if $A$ is a $\mathrm{GE}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $Γ(A)$, we prove that $Y(A)$ is simply connected if and only if $A$ is universal for $\mathrm{GE}_2$. More precisely, our main theorem is that for any commutative ring $A$ the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.

math.KT↗

arXiv2020

The third homology of $\mathrm{SL}_2(\mathbb{Q})$

We calculate the third homology of $\mathrm{SL}_2(\mathbb{Q})$ with half-integral coefficients. Corresponding to each prime $p$ there is an operator on this group with square the identity. The kernel of the (split surjective) homomorphism to the indecomposable $K_3$ of $\mathbb{Q}$ is a the direct sum over all primes of the $(-1)$-eigenspaces of these operators. The $(-1)$-eigenspace of the operator corresponding to the prime $p$ is cyclic of order the odd part of $p+1$.

math.KT↗

arXiv2020

The homology of $\mathrm{SL}_2$ of discrete valuation rings

Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}]) \to H_3(\mathrm{SL}_2(F),\mathbb{Z}[\frac{1}{2}])\to \mathcal{RP}_1(k)[\frac{1}{2}]\to 0$, where $\mathcal{RP}_1(k)$ is the refined scissors congruence group of $k$. Let $Γ_0(\mathfrak{m}_A)$ denote the congruence subgroup consisting of matrices in $\mathrm{SL}_2(A)$ whose lower off-diagonal entry lies in the maximal ideal $\mathfrak{m}_A$. We also prove that there is an exact sequence $0\to \overline{\mathcal{P}}(k)[\frac{1}{2}]\to H_2(Γ_0(\mathfrak{m}_A),\mathbb{Z}[\frac{1}{2}])\to H_2(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}])\to I^2(k)[\frac{1}{2}]\to 0$, where $I^2(k)$ is the second power of the fundamental ideal of the Grothendieck-Witt ring $\mathrm{GW}(k)$ and $\overline{\mathcal{P}}(k)$ is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) $\mathcal{P}(k)$ of $k$.

math.KT↗

arXiv2018

Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system

In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the energy exchange between normal modes. In the weakly nonlinear regime, those interactions may facilitate energy equipartition in Fourier space. We consider analytically resonant wave-wave interactions for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) system. Using a number-theoretical approach based on cyclotomic polynomials, we show that the problem of finding exact resonances for a system of N particles is equivalent to a Diophantine equation whose solutions depend sensitively on the set of divisors of N. We provide an algorithm to construct all possible resonances, based on two methods: pairing-off and cyclotomic, which we introduce to build up explicit solutions to the 4-, 5- and 6-wave resonant conditions. Our results shed some light in the understanding of the long-standing FPUT paradox, regarding the sensitivity of the resonant manifolds with respect to the number of particles N and the corresponding time scale of the interactions leading to thermalisation. In this light we demonstrate that 6-wave resonances always exist for any N, while 5-wave resonances exist if N is divisible by 3 and N > 6. It is known (for finite N) that 4-wave resonances do not mix energy across the spectrum, so we investigate whether 5-wave resonances can produce energy mixing across a significant region of the Fourier spectrum by analysing the interconnected network of Fourier modes that can interact nonlinearly via resonances. The answer depends on the set of odd divisors of N that are not divisible by 3: the size of this set determines the number of dynamically independent components, corresponding to independent constants of motion (energies). We show that 6-wave resonances connect all these independent components, providing in principle a restoring mechanism for full-scale thermalisation.

nlin.CD↗

arXiv2017

Tate kernels, etale K-theory and the Gross kernel

For an odd prime $p$ and a number field $F$ containing a $p$th root of unity, we study generalised Tate kernels, $D_F^{[i,n]}$, for $i\in \mathbb{Z}$ and $n\geq 1$, having the properties that if $i\geq 2$ and if either $p$ does not divide $i$ or $μ_{p^n}\subset F$ then there are natural isomorphisms $D_F^{[i,n]}\cong K^{\mbox{\tiny ét}}_{2i-1}(O_F^S)/p^n$, and that they are periodic modulo a power of $p$ which depends on $F$ and $n$. Our main result is that if the Gross-Jaulent conjecture holds for $(F,p)$ then there is a natural isomorphism $D_F^{[i,n]}\cong\mathcal{E}_F/p^n$ where $\mathcal{E}_F$ is the Gross kernel. We apply this result to compute lower bounds for capitulation kernels in even étale $K$-theory.

math.NT↗

arXiv2016

The third homology of SL_2 of local rings

We describe the third homology of SL_2 of local rings over Z[1/2] in terms of a refined Bloch group. We use this to derive a localization sequence for the third homology of SL_2 of certain discrete valuation rings, and to make explicit computations for p-adic integers and other local rings.

math.KT↗

arXiv2016

The third homology of SL_2 of fields with discrete valuation

For a field F with discrete valuation and residue field $k$ we relate the third homology of SL_2(F) with half-integral coefficients to the third homology of SL_2(k) and a certain refined scissors congruence group of k. As an application, we obtain explicit calculations of the third homology of SL_2 of certain higher-dimensional local fields in terms of scissors congruence groups, in the sense of Dupont and Sah, of the residue fields of the associated valuations.

math.KT↗

arXiv2015

The second homology of SL_2 of S-integers

We calculate the structure of the finitely-generated groups H_2(SL_2(Z[1/m])) when m is a multiple of 6. We construct explicit homology classes which generate these groups and have prescribed orders. When n is at least 2 and m is the product of the first n primes, we combine our results with those of Jun Morita to deduce that the projection St(2, Z[1/m]) --> SL_2(Z[1/m]) is a universal central extension, where St(2,-) is the rank one Steinberg group. The main structure theorem applies more generally to rings of S-integers with sufficiently many units. For a wide class of rings, the explicit homology classes we construct map to symbols in rank one K_2 of the ring.

math.KT↗

arXiv2014

On third homology of SL_2 and weak homotopy invariance

The goal of the paper is to achieve - in the special case of the linear group SL_2 - some understanding of the relation between group homology and its A^1-invariant replacement. We discuss some of the general properties of A^1-invariant group homology, such as stabilization sequences and Grothendieck-Witt module structures. Together with very precise knowledge about refined Bloch groups, these methods allow to deduce that in general there is a rather large difference between group homology and its A^1-invariant version. In other words, weak homotopy invariance fails for SL_2 over many families of non-algebraically closed fields.

math.KT↗

arXiv2014

Low-dimensional Homology of SL_2(k[t,1/t])

We prove analogues of the fundamental theorem of algebraic K-theory for the second and third homology of SL_2 over an infinite field k. The statements involve Milnor-Witt K-theory and scissors congruence groups. We use these results to calculate the low-dimensional homology of SL_2 of Laurent polynomials over certain fields.

math.KT↗

arXiv2013

A refined Bloch group and the third homology of SL_2 of a field

We use the properties of the refined Bloch group of a field to prove that H_3 of SL_2 of a global field is never finitely generated, and to calculate - up to some 2-torsion - H_3 of SL_2 of local fields with finite residue field of odd characteristic. We also give lower bounds for the 3-torsion in the H_3 of SL_2 of rings of S-integers.

math.KT↗

arXiv2013

A Bloch-Wigner complex for SL_2

We introduce a refinement of the Bloch-Wigner complex of a field F. This is a complex of modules over the multiplicative group of the field. Instead of computing K_2 and indecomposable K_3 - as the classical Bloch-Wigner complex does - it calculates the second and third integral homology of SL_2 of the field. On passing to coinvariants for the action of the multiplicative group we recover the classical Bloch-Wigner complex. The case of finite fields is included throughout the article.

math.KT↗

arXiv2011

Lines crossing a tetrahedron and the Bloch group

We consider a simple modification of the Chow group CH^2(Spec(k),3) using only linear subvarieties in affine spaces and show that it maps surjectively to the Bloch group B(k) for any infinite field k. We also describe the kernel of this map.

math.NT↗

arXiv2009

Homology stability for the special linear group of a field and Milnor-Witt K-theory

Let F be a field of characteristic zero and let f(t,n) be the stabilization homomorphism from the n-th integral homology of SL(t,F) to the n-th homology of SL(t+1,F). We prove the following results: For all n, f(t,n) is an isomorphism if t is at least n+1, and is surjective for t=n, confirming a conjecture of C-H. Sah. Furthermore if n is odd, then f(n,n) is an isomorphism and when n is even the kernel of f(n,n) is the (n+1)st power of the fundamental ideal of the Witt Ring of the field.. If n is even, then the cokernel of f(n-1,n) is naturally isomorphic to the n-th Milnor-Witt K-group of F, MWK(n,F) and when n>2 is odd the cokernel of f(n-1,n) is the square of the nth Milnor K-group of F.

math.KT↗

arXiv2008

The third homology of the special linear group of a field

We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory.

math.KT↗