Related papers: On the extension problems for three 33-stem homoto…

Groups $\Pi_k(X;\sigma)$ of "flagged homotopies" are introduced of which the usual (abelian for $k>1$) homotopy groups $\pi_k(X;p)$ is the limit case for flags $\sigma$ contracted to a point $p$. Calculus of exterior forms with values in…

Let $\ell$ be a commutative ring with unit. To every pair of $\ell$-algebras $A$ and $B$ one can associate a simplicial set $\hom(A,B^\Delta)$ so that $\pi_0\hom(A,B^\Delta)$ equals the set of polynomial homotopy classes of morphisms from…

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The extension problem in Torelli topology is the problem of determining when a diffeomorphism of compact…

For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable…

The Kervaire Problem is an unsolved problem in Stable Homotopy Theory. The first interesting example is in dimension $30$. There exists a closed stably-parallelizable manifold $\tilde{M}^{30}$ with Arf-Kervaire invariant 1. It is unknown,…

One of the central problems in the topology of hyperplane arrangements is determining whether the complement is a $K(\pi,1)$-space. In this paper, we study Manin--Schechtman arrangements, introduced as higher-dimensional analogs of the…

We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two…

Extending our method for investigating Real cobordism (which was recently used by Hill, Hopkins and Ravenel in their solution of the Kervaire invariant 1 problem), we investigate the $RO(G)$-graded homotopy groups of a (non-complete)…

This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the…

We discuss four off-shell N=4 D=1 supersymmetry transformations, their associated one-dimensional sigma-models and their mutual relations. They are given by I) the (4,4)_{lin} linear supermultiplet (supersymmetric extension of R^4), II) the…

Let $M$ be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If $M$ has nonvanishing second homotopy group, then it is known that the $\pi_2$-systole of $M$ (i.e. the minimal achievable area of homotopically…

This paper is motivated by recent developments in group stability, high dimensional expansion, local testability of error correcting codes and topological property testing. In Part I, we formulate and motivate three stability problems: 1.…

This document contains large-format Adams charts that compute 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. The charts are essentially complete through the 61-stem and contain partial…

We give the first tractable and systematic examples of nontrivial higher digraph homotopy groups. To do this we define relative digraph homotopy groups and show these satisfy a long exact sequence analogous to the relative homotopy groups…

This is a selection of open problems dealing with ``large'' (non locally compact) topological groups and concerning extreme amenability (fixed point on compacta property), oscillation stability, universal minimal flows and other aspects of…

We construct homotopically non-trivial maps from S^m to S^n with arbitrarily small 3-dilation for certain pairs (m,n). The simplest example is m=4, n=3. Other examples include arbitrarily large values of m and n. We show that a homotopy…

We provide a mathematical realization of a conjecture by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an $\Omega$-spectrum $\mathit{IP}_*$…

We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, SU(3), Sp(2), and G_2. We use the cell structures of these Lie groups and the standard methods of homotopy theory.

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $\Sigma^2M$ as wedge sums of some elementary…

We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the…