Martin Palmer

Institutul de Matematică Simion Stoilow al Academiei Române

We prove homological stability for sequences of "oriented configuration spaces" as the number of points in the configuration goes to infinity. These are spaces of configurations of n points in a connected manifold M of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an orientation: an ordering of th...

We introduce a new map between configuration spaces of points in a background manifold - the replication map - and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the background manifold is closed, in which case the classical stabilisation map does not exist. We then establish cond...

The purpose of this note is to clarify some details in McDuff and Segal's proof of the group-completion theorem in McDuff and Segal [Homology fibrations and the ‘group-completion’ theorem, Invent. Math. 31 (1975/76), 279–284] and generalize this and the homology fibration criterion of McDuff [Configuration spaces of positive and negative particles,...

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary d...

We prove that the mapping class group of the one‐holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once‐punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of in...

We prove that, for any infinite-type surface S , the integral homology of the closure of the compactly-supported mapping class group \smash{\overline{\mathrm{PMap}_c(S)}} and of the Torelli group \mathcal{T}(S) is uncountable in every positive degree. By our results in [arXiv:2211.07470] and other known computations, such a statement cannot be true...

We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles. The former are Gibbons–Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monop...

In previous work we constructed twisted representations of mapping class groups of surfaces, depending on a choice of representation $V$ of the Heisenberg group $\mathcal{H}$. For certain $V$ we were able to untwist these mapping class group representations. Here, we study the restrictions of our twisted representations to different subgroups of th...

A wide family of homological representations of surface braid groups and mapping class groups of surfaces was developed in arXiv:1910.13423. These representations are naturally defined as functors on a category whose automorphism groups are the family of groups under consideration, and whose richer structure may be used to prove twisted homological...

We prove that some of the classical homological stability results for configuration spaces of points in manifolds can be lifted to motivic cohomology.

We prove that, for any infinite-type surface $S$, the integral homology of the pure mapping class group $\mathrm{PMap}(S)$ and of the Torelli group $\mathcal{T}(S)$ is uncountable in every positive degree. By our results in arxiv:2211.07470 and other known computations, such a statement cannot be true for the full mapping class group $\mathrm{Map}(...

We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles. The former are Gibbons-Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monop...

We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of in...

We construct a 3-variable enrichment of the Lawrence-Krammer-Bigelow (LKB) representation of the braid groups, which is the limit of a pro-nilpotent tower of representations having the original LKB representation as its bottom layer. We also construct analogous pro-nilpotent towers of representations of surface braid groups and loop braid groups.

Given a manifold M and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of M to the group of isotopy classes of diffeomorphisms of M that fix the basepoint. This map is well-studied in dimension d=2 and is part of the Birman ex...

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. T...

Understanding the lower central series of a group is, in general, a difficult task. It is, however, a rewarding one: computing the lower central series and the associated Lie algebras of a group or of some of its subgroups can lead to a deep understanding of the underlying structure of that group. Our goal here is to showcase several techniques aim...

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. T...

For a given bundle \(\xi :E \rightarrow M\) over a manifold, configuration-section spaces parametrise finite subsets \(z \subseteq M\) equipped with a section of \(\xi \) defined on \(M \smallsetminus z\), with prescribed “charge” in a neighbourhood of the points z. These spaces may be interpreted physically as spaces of fields that are permitted t...

We study the action of the mapping class group of $\Sigma = \Sigma_{g,1}$ on the homology of configuration spaces with coefficients twisted by the discrete Heisenberg group $\mathcal{H} = \mathcal{H}(\Sigma)$, or more generally by any representation $V$ of $\mathcal{H}$. In general, this is a twisted representation of the mapping class group $\math...

A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to moduli spaces of submanifolds of higher dimension, where stability is with respect to the number of components having a fixed diffeomorphism type and iso...

We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general account of the fundamental relationships (non-degenerate pairings, embeddings, isomorphisms) between the many diff...

We prove that some of the classical homological stability results for configuration spaces of points in manifolds can be lifted to motivic cohomology.

Given a manifold M and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of M to the group of isotopy classes of diffeomorphisms of M that fix the basepoint. This map is well-studied in dimension d=2 and is part of the Birman ex...

For a given bundle $\xi \colon E \to M$ over a manifold, configuration-section spaces on $\xi$ parametrise finite subsets $z \subseteq M$ equipped with a section of $\xi$ defined on $M \smallsetminus z$, with prescribed "charge" in a neighbourhood of the points $z$. These spaces may be interpreted physically as spaces of fields that are permitted t...

The families of braid groups, surface braid groups, mapping class groups and loop braid groups have a representation theory of "wild type", so it is very useful to be able to construct such representations topologically, so that they may be understood by topological or geometric methods. For the braid groups $\mathbf{B}_n$, Lawrence and Bigelow hav...

Groups of a topological origin, such as braid groups and mapping class groups, often have "wild" representation theory. One would therefore like to construct representations of these groups topologically, in order to use topological tools to study them. A key example is Bigelow and Krammer's proof of the linearity of the braid groups, which uses to...

The homology of configuration spaces of point-particles in manifolds has been studied intensively since the 1970s; in particular it is known to be stable if the underlying manifold is connected and open. Closely related to configuration spaces are moduli spaces of manifolds with marked points, and in [Tillmann, 2016] this relation was used to show...

A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to moduli spaces of submanifolds of higher dimension, where stability is with respect to the number of components having a fixed diffeomorphism type and iso...

LetM be an open, connected manifold. A classical theorem of McDuffand Segal states that the sequence (Cn(M)) of configuration spaces of n unordered, distinct points in M is homologically stable with coefficients in ℤ -in each degree, the integral homology is eventually independent of n. The purpose of this paper is to prove that this phenomenon als...

In the first part of this note, we review and compare various instances of the notion of twisted coefficient system, a.k.a. polynomial functor, appearing in the literature. This notion hinges on how one defines the degree of a functor from C to an abelian category, for various different structures on C. In the second part, we focus on twisted coeff...

In [Pal13] the second author proved that the sequence of "oriented" configuration spaces on an open connected manifold exhibits homological stability as the number of particles goes to infinity. To complement that result we identify the corresponding limiting space, up to homology equivalence, as a certain explicit double cover of a section space....

Fix a connected open manifold M and a path-connected space X. Then the sequence C_n(M,X) of configuration spaces of n distinct unordered points in M equipped with labels from X is known to be homologically stable: in each degree, the integral homology is eventually independent of n. In this note we prove that this is also true for homology with twi...

In [Pal13] (arXiv:1106.4540) the second author proved that "oriented" configuration spaces exhibit homological stability. To complement that result we identify the limiting space, up to homology equivalence, as a certain explicit double cover of a section space. Along the way we also prove that the scanning map of McDuff in [McD75] for unordered co...