arXiv subjects
Publications and source records attributed to Rachael Boyd.
We survey computations and tools concerning the mapping class group of a compact, oriented, connected 3-manifold $M$. We provide a guide to the literature and sketch proofs for various families of irreducible and geometric 3-manifolds. We also consider JSJ and prime decompositions of 3-manifolds, and consequences for their mapping class groups.
We study the moduli space $B\textrm{Diff}^+(M)$, for $M$ a reducible, oriented 3-manifold with irreducible prime factors $P_1,\ldots,P_n$. A programme of César de Sá-Rourke, Hendriks-Laudenbach, and Hendriks-McCullough studies the homotopy type of $\textrm{Diff}^+(M)$ in terms of the $\textrm{Diff}^+(P_i)$. Inspired by a delooping proposed by Hatcher, we construct a map from $B\textrm{Diff}^+(M)$ to $B\textrm{Diff}^+(P_1 \sqcup \dots \sqcup P_n)$, called the splitting map, that yields a prime decomposition fibre sequence. The fibre $H_g(P_1, \dots, P_n)$ is a space of $1$-handle attachments which we describe geometrically as a homotopy colimit of certain configuration spaces on the $P_i$. Firstly, this allows us to show that for $n>0$ the fibre is equivalent to a finite, connected cell complex. Secondly, this makes the fibre sequence an effective tool for computations, which we illustrate by computing the rational cohomology ring of $B\textrm{Diff}^+\!\left((S^1\times S^2)^{\sharp 2}\right)$.
We give a brief introduction to the geometric and combinatorial group theory of Artin groups. In particular we introduce the $K(π,1)$ conjecture for Artin groups and survey known results as of January 2024. These notes were written as companion notes for the MFO mini-workshop 2405a "Artin groups meet triangulated categories" alongside Edmund Heng's notes "Introduction to stability conditions and its relation to the $K(π,1)$ conjecture for Artin groups".
We show that the homology of a Temperley-Lieb algebra on an even number of strands has a rich algebraic structure and is highly nontrivial in general. This is achieved by proving that it is entirely governed by a differential graded algebra: the differential graded algebra of planar loops. We provide a small model for this dga, and use it to obtain consequences on homology.
We study the unparametrised smooth embedding space of a Hopf link in $\mathbb{R}^3$, and prove that it is homotopy equivalent to the closed 3-manifold $S^3/\mathbb{Q}_8$. As an intermediate step in the proof, we show that the inclusion of the subspace of round embeddings is a homotopy equivalence. We provide analogous results for the unparametrised smooth embedding space of a Hopf link in $S^3$, which we show is homotopy equivalent to $\mathbb{R} P^2\times \mathbb{R} P^2$.
Let $H_g$ denote the 4-dimensional handlebody of genus $g$ and $U_g$ its boundary. We show that for all $g \ge 0$ the map from $B Homeo(H_g)$ to $B Homeo(U_g)$ induced by restriction to the boundary admits a section.
We study the homotopy type of the space $E(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Our main result is a simple description of the fundamental group, or motion group, of $E(L)$, and we extend this to a description of the motion group of embeddings in $S^3$. The main tool we build is a semi-simplicial space of separating systems, which we show is homotopy equivalent to $E(L)$. This combinatorial object provides a gateway to studying the homotopy type of $E(L)$ via the homotopy type of the spaces $E(L_i)$.
We study the classifying space B Diff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. In the case that M is reducible we build a contractible space parametrising the systems of reducing spheres. We use this to prove that if M has non-empty boundary, then B Diff(M rel boundary) has the homotopy type of a finite CW complex. This was conjectured by Kontsevich and appears on the Kirby problem list as Problem 3.48. As a consequence, we are able to show that for every compact, orientable 3-manifold M, B Diff(M) has finite type.
We show that the homology of the partition algebras, interpreted as appropriate Tor-groups, is isomorphic to that of the symmetric groups in a range of degrees that increases with the number of nodes. Furthermore, we show that when the defining parameter $δ$ of the partition algebra is invertible, the homology of the partition algebra is in fact isomorphic to the homology of the symmetric group in all degrees. These results parallel those obtained for the Brauer algebras in the authors' earlier work, but with significant differences and difficulties in the inductive resolution and high acyclicity arguments required to prove them. Our results join the growing literature on homological stability for algebras, which now encompasses the Temperley-Lieb, Brauer and partition algebras, as well as the Iwahori-Hecke algebras of types A and B.
In this paper we investigate properties of the Artin monoid Cayley graph. This is the Cayley graph of an Artin group $A_Γ$ with respect to the (infinite) generating set given by the associated Artin monoid $A^+_Γ$. In a previous paper, the first three authors introduced a monoid Deligne complex and showed that this complex is contractible for all Artin groups. In this paper, we show that the Artin monoid Cayley graph is quasi-isometric to a modification of the Deligne complex for $A_Γ$ obtained by coning off translates of the monoid Deligne complex. We then address the question of when the monoid Cayley graph has infinite diameter. We conjecture that this holds for all Artin groups of infinite type. We give a set of criteria that imply infinite diameter, and using existing solutions to the word problem for large-type Artin groups and 3-free Artin groups, we prove that the conjecture holds for any Artin group containing a 3-generator subgroup of one of these two types.
We investigate the minimal genus problem for the second homology of a right angled Artin group (RAAG). Firstly, we present a lower bound for the minimal genus of a second homology class, equal to half the rank of the corresponding cap product matrix. We show that for complete graphs, trees, and complete bipartite graphs, this bound is an equality, and furthermore in these cases the minimal genus can always be realised by a disjoint union of tori. Additionally, we give a full characterisation of classes that are representable by a single torus. However, the minimal genus of a second homology class of a RAAG is not always realised by a disjoint union of tori as an example we construct in the pentagon shows.
This paper studies the homology and cohomology of the Temperley-Lieb algebra TL_n(a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a=v+v^{-1} for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n-2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even. Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of 'planar injective words' that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL_n(a) is not flat over TL_m(a) for m<n, so that Shapiro's lemma is unavailable. We resolve this difficulty by constructing what we call 'inductive resolutions' of the relevant modules. Vanishing results for the homology and cohomology of Temperley-Lieb algebras can also be obtained from the existence of the Jones-Wenzl projector. Our own vanishing results are in general far stronger than these, but in a restricted case we are able to obtain additional vanishing results via existence of the Jones-Wenzl projector. We believe that these results, together with the second author's work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.
This paper investigates the homology of the Brauer algebras, interpreted as appropriate Tor-groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when the parameter is not invertible, this isomorphism still holds in a range of degrees that increases with n.
In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(π,1) conjecture for these groups. Using a notion of Artin monoid cosets, we construct a version of the Deligne complex for Artin monoids. We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph.
This paper studies combinatorial properties of the 'complex of planar injective words', a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number. We obtain an alternating sum formula for the representation given by its top-dimensional homology module and, under further restrictions on the ground ring, we decompose this module in terms of certain standard Young tableaux. This trio of results - inspired by results of Reiner and Webb for the complex of injective words - can be viewed as an interpretation of the n-th Fine number as the 'planar' or 'Dyck path' analogue of the number of derangements of n letters. This interpretation has precursors in the literature, but here emerges naturally from considerations in homological stability. Our final result shows a surprising connection between the boundary maps of our complex and the Jacobsthal numbers.
We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.
We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the $K(π,1)$ conjecture holds for the associated family of Artin groups this establishes homological stability for these groups. In particular, this recovers and extends Arnol'd's proof of stability for the Artin groups of type $A$, $B$ and $D$.