具有有限阿贝尔2生成元基本群的4维流形的同伦分类

图示、球面同伦群以及长链环与纽结空间

Graphing, homotopy groups of spheres, and spaces of long links and knots

本文研究高余维长链环空间的同伦群，利用图示映射证明球面同伦群在广义无三重点范围内构成全部来源，并描述范围外长纽结首个非平凡同伦群的 Borromean 环与 Hopf 纤维化生成元。

The homotopy types of free racks and quandles

We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs of Milnor's theorem on free groups for these theories and their pointed variants, identifying the homotopy types of the free racks and free quandles on spaces of generators. These results allow us to complete the stable classification of racks and quandles by identifying the ring spectra that model their stable homotopy theories. As an application, we show that the stable homotopy of a knot quandle is, in general, mor…

Towards a nonabelian cohomology of forms

We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie superalgebra H*(M,G) we obtain numerical smooth invariants--as opposed to homotopy invariants--for manifolds. The corresponding Hodge theory yields finiteness of nonabelian Betti numbers. A genralized Poincaré lemma, along with a Poincaré duality, a Mayer-Vietoris, and a particularly empowered Bockstein makes our cohomology computable. Bockstein also allows u…

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds II. Invariants of 3-manifolds with $b_1=1$

In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $ω$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,ω)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $ω$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.

Diagram groups and directed 2-complexes: homotopy and homology

We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so called rigidity of the R.Thompson's group $F$ and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F_\infty) each of which contain…