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Graham Ellis

Publications and source records attributed to Graham Ellis.

7 recordsLinked to original sources
arXiv2020

Introductory computations in the cohomology of arithmetic groups

This paper describes an approach to computer aided calculations in the cohomology of arithmetic groups. It complements existing literature on the topic by emphasizing homotopies and perturbation techniques, rather than cellular subdivision, as the tools for implementing on a computer topological constructions that fail to preserve cellular structures Furthermore, it focuses on calculating integral cohomology rather than just rational cohomology or cohomology at large primes. In particular, the paper describes and fully implements algorithms for computing Hecke operators on the integral cuspidal cohomology of congruence subgroups $\Gamma$ of $SL_2(\mathbb Z)$, and then partially implements versions of the algorithms for the special linear group $SL_2({\cal O}_d)$ over various rings of quadratic integers ${\cal O}_d$. The approach is also relevant for computations on congruence subgroups of $SL_m({\cal O}_d)$, $m\ge 2$.

math.NT↗
arXiv2019

Cohomology with local coefficients and knotted manifolds

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings $N\hookrightarrow M$ of one topological manifold into another. More specifically, we describe an algorithm for computing the homology $H_n(X,A)$ and cohomology $H^n(X,A)$ of a finite connected CW-complex X with local coefficients in a $\mathbb Z\pi_1X$-module $A$ when $A$ is finitely generated over $\mathbb Z$. It can be used, in particular, to compute the integral cohomology $H^n(\widetilde X_H,\mathbb Z)$ and induced homomorphism $H^n(X,\mathbb Z) \rightarrow H^n(\widetilde X_H,\mathbb Z)$ for the covering map $p\colon \widetilde X_H \rightarrow X$ associated to a finite index subgroup $H < \pi_1X$, as well as the corresponding homology homomorphism. We illustrate an open-source implementation of the algorithm by using it to show that: (i) the degree $2$ homology group $H_2(\widetilde X_H,\mathbb Z)$ distinguishes between the homotopy types of the complements $X\subset \mathbb R^4$ of the spun Hopf link and Satoh's tube map of the welded Hopf link (these two complements having isomorphic fundamental groups and integral homology); (ii) the degree $1$ homology homomorphism $H_1(p^{-1}(B),\mathbb Z) \rightarrow H_1(\widetilde X_H,\mathbb Z)$ distinguishes between the homeomorphism types of the complements $X\subset \mathbb R^3$ of the granny knot and the reef knot, where $B\subset X$ is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology).

math.AT↗
arXiv2011

On the integral homology of PSL4(Z) and other arithmetic groups

We determine the integral homology of PSL4(Z) in degrees at most 5 and determine its p-part in higher degrees for the primes p>=5. Our method applies to other arithmetic groups; as illustrations we include descriptions of the integral homology of PGL3(Z[i]) and PGL3(Z[\exp(2\pi{i}/3)]) in degrees at most 5.

math.KT↗
arXiv2010

Persistent homology of groups

We introduce and investigate notions of persistent homology for p-groups and for coclass trees of p-groups. Using computer techniques we show that persistent homology provides fairly strong homological invariants for p-groups of order at most 81. The strength of these invariants, and some elementary theoretical properties, suggest that persistent homology may be a useful tool in the study of prime-power groups.

math.GR↗
arXiv2009

Wythoff polytopes and low-dimensional homology of Mathieu groups

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as $H_5(M_{23},\ZZ)=\ZZ_7$ and $H_3(M_{24},\ZZ)=\ZZ_{12}$. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free $\ZZ M_n$-resolution. Both methods apply in principle to arbitrary finite groups.

math.GR↗
arXiv2008

A colimit of classifying spaces

We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu's group-theoretic description of the homotopy groups of a 2-sphere.

math.GR↗