Related papers: On a homotopy relation between the 2-local geometr…

We study the homotopy relation between the standard 2-local geometry and the Bouc complex for the sporadic group Co3. We also give a result concerning the relative projectivity of the reduced Lefschetz module associated to the aformentioned…

We investigate the homotopy type of a certain homogeneous space for a simple complex algebraic group. We calculate some of its classical topological invariants and introduce a new one. We also propose several conjectures about its…

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component…

We give an elementary proof of the Hurewicz theorem relating homotopy and homology groups of a cubical Kan complex. Our approach is based on the notion of a loop space of a cubical set, developed in a companion paper ``Homotopy groups of…

We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence…

There is a well-established procedure of assigning a strong homotopy Lie algebra of local observables to a multisymplectic manifold which can be regarded as part of a categorified Poisson structure. For a 2-plectic manifold, the resulting…

We explore the relations among quadratic modules, 2-crossed modules, crossed squares and simplicial groups with Moore complex of length 2.

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse…

A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to…

In this paper we compute the mod 2 cohomology of the McLaughlin group, which is one of the sporadic simple groups.

We determine the 2-local homotopy types of G_2-gauge groups over S^4.

We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy…

Two locally generic maps f,g : M^n --> R^{2n-1} are regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if n is not 3 and M^n is a closed n-manifold then the regular…

We address the (pointed) homotopy theory of 2-crossed modules (of groups), which are known to faithfully represent Gray 3-groupoids, with a single object, and also connected homotopy 3-types. The homotopy relation between 2-crossed module…

We consider the loci of d-elliptic curves in $M_2$, and corresponding loci of d-elliptic surfaces in $A_2$. We show how a description of these loci as quotients of a product of modular curves can be used to calculate cohomology of natural…

This article presents a 3-local characterisation of the sporadic simple group McL and its automorphism group. The theorem is underpinned by a further identification theorem the proof of which is character theoretic. The main theorem is…

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and…

Let $M$ be an orientable, simply-connected, closed, non-spin 4-manifold and let $\mathcal{G}_k(M)$ be the gauge group of the principal $G$-bundle over $M$ with second Chern class $k\in\mathbb{Z}$. It is known that the homotopy type of…

We identify the sporadic simple group $\mathrm{M}_{12}$ and the simple group $\mathrm{SL}_3(3)$ from some part of their 3-local structure and give a graph theoretic analogue of the resulting theorem.

We show that for an oriented 4-dimensional Poincar\'e complex with finite fundamental group, whose 2-Sylow subgroup is abelian with at most 2 generators, the homotopy type is determined by its quadratic 2-type.