arXiv subjects
Publications and source records attributed to Jacob Mostovoy.
We find the complete rational homology for the finite subset spaces of a $d$-dimensional sphere. We also determine the integral homology in top $d$ degrees and obtain a partial description of it in codimension $d$.
We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.
We study the space $Q_n$ of all configurations of $n$ ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for $n<6$ and describe its homology for $n=6,7$. For arbitrary $n$, we compute its first homology and its Euler characteristic. We use three geometric approaches. On one hand, $Q_n$ is naturally defined as the complement to an arrangement of codimension-2 subtori in a real torus. On the other hand, $Q_n$ is homotopy equivalent to an explicit nonpositively curved cubical complex. Finally, $Q_n$ can also be assembled from no-3-equal manifolds of the real line. We also observe that, up to homotopy, $Q_n$ may be identified with a subspace of the oriented double cover of the moduli space $\overline{\mathcal{M}}_{0,n}(\mathbb{R})$ of stable real rational curves with $n$ marked points. This gives an embedding of $\pi_1 Q_n$ into the pure cactus group. As a corollary, we see that $\pi_1 Q_n$ is residually nilpotent.
We define a homology theory for pre-crossed modules that specifies to rack homology in the case when the pre-crossed module is freely generated by a rack.
Cactus doodles are combinatorial/geometric objects that are related to cactus groups in the same way as knots are related to braids. We define them in terms of local moves on plane curves, show that they can be obtained from elements of the cactus group by a "closing" procedure and establish some of their basic properties.
We describe the construction of a minimal presentation for the group of planar pure braids $\overline{P}_n$ on $n$ strands. The generators of this presentation are dual to the generators of the cohomology ring of $\overline{P}_n$ found by Y.Baryshnikov while the relations lie in the commutator subgroup of $\overline{P}_n$. We also construct an explicit (although non-minimal) presentation for the pure cactus group.
We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather than out of crossings. We describe an equivalence relation on clasp diagrams which produces the isotopy classes of knots as equivalence classes. This equivalence relation is generated by local moves similar to the Reidemeister moves. Clasp diagrams produce particularly simple Seifert surfaces for knots and lead to an explicit formula for the Alexander-Conway polynomial. They are also well-suited for the study of the Vassiliev invariants; we show that any such invariant can be obtained via subdiagram count in the clasp diagrams.
In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley-Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in arXiv:1904.00121 [math.KT].
In this, largely expository, note, we show how the simplicial structure of the moduli spaces of stable rational curves with marked points allows to produce explicit equations for these spaces. The key argument is an elementary combinatorial statement about the sets of trees with marked leaves.
The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(\mathbb{R})$ of $n$ labelled points in $\mathbb{R}$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.
We show that the pure cactus group $Γ_{n+1}$ is residually nilpotent and exhibit a surjective homomorphism $Γ_{n+1} \to (\mathbb{Z}/2\mathbb{Z})^{2^n-n(n+1)/2-1}$ whose kernel is residually torsion-free nilpotent.
We interpret augmented racks as a certain kind of multiplicative graphs and show that this point of view is natural for defining rack homology. We also define the analogue of the group algebra for these objects; in particular, we see how discrete racks give rise to Hopf algebras and Lie algebras in the Loday-Pirashvili category $\mathcal{LM}$. Finally, we discuss the integration of Lie algebras in $\mathcal{LM}$ in the context of multiplicative graphs and augmented racks.
Dold-Thom functors are generalizations of infinite symmetric products, where integer multiplicities of points are replaced by composable elements of a partial abelian monoid. It is well-known that for any connective homology theory, the machinery of $Γ$-spaces produces the corresponding linear Dold-Thom functor. In this note we construct such functors directly from spectra by exhibiting a partial monoid corresponding to a spectrum.
In this paper we study the space of morphisms from a complex projective space to a compact smooth toric variety X. It is shown that the first author's stability theorem for the spaces of rational maps from CP^m to CP^n extends to the spaces of continuous morphisms from CP^m to X, essentially, with the same proof. In the case of curves, our result improves the known bounds for the stabilization dimension.
Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and prove that it is obtained by localizing LM with respect to adding a "small twist".
We show that for any n real periodic functions f_1,..., f_n with the same period, such that f_i>0 for i 0, there is a closed curve in R^{n+1} with curvatures k_1, ..., k_n such that |k_i(t)-f_i(t)| < e for all i and t. This neither holds for closed curves in the hyperbolic space H^{n+1}, nor for parametric families of closed curves in R^{n+1}.
We show that for an m-connected cell complex X the space exp_k X of non-empty subsets of X of cardinality at most k is (m + k - 2)-connected
In this note we point out that the definition of the universal enveloping dialgebra for a Leibniz algebra is consistent with the interpretation of a Leibniz algebra as a generalization not of a Lie algebra, but of the adjoint representation of a Lie algebra. From this point of view, the formal integration problem of Leibniz algebras is, essentially, trivial.