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

We give new polynomial-time algorithms for testing isomorphism of a class of groups given by multiplication tables (GpI). Two results (Cannon & Holt, J. Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces to the…

We determine $\pi_*(BDiff_\partial(D^{2n})) \otimes \mathbb{Q}$ for $2n \geq 6$ completely in degrees $* \leq 4n-10$, far beyond the pseudoisotopy stable range. Furthermore, above these degrees we discover a systematic structure in these…

The homotopy groups of a space are endowed with homotopy operations which define the \Pi-algebra of the space. An Eilenberg-MacLane space is the realization of a \Pi-algebra concentrated in one degree. In this paper, we provide necessary…

We compute the spoke-graded $C_3$-equivariant stable homotopy groups of spheres $\pi_{i, j}^{C_3}$, for stems less than 25 (i.e. $i\leq 25$) and for weights between -16 and 16 (i.e. $-16\leq j\leq 16$). In particular, for $j=2k$, this…

This paper addresses several isotopy problems on $4$-manifolds. First, we classify the isotopy classes of embeddings of $\Sigma$ in $\Sigma\times S^2$ that are geometrically dual to $\{\mbox{pt}\}\times S^2$, where $\Sigma$ is a closed…

The homotopy group $\pi_{n-k} ({\bf C}^{n+1}-V)$ where $V$ is a hypersurface with a singular locus of dimension $k$ and good behavior at infinity is described using generic pencils. This is analogous to the van Kampen procedure for finding…

We consider the semigroup $Ext(A,B)$ of extensions of a separable C*-algebra $A$ by a stable C*-algebra $B$ modulo unitary equivalence and modulo asymptotically split extensions. This semigroup contains the group $Ext^{-1/2}(A,B)$ of…

Let S be a site. First we define the 3-category of torsors under a Picard S-2-stack and we compute its homotopy groups. Using calculus of fractions we define also a pure algebraic analogue of the 3-category of torsors under a Picard…

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $5$-manifolds. As applications, we compute the reduced $K$-groups of $M$ and show that the suspension map between…

In a recent paper, it was shown that the problem of existence of a continuous map $X \to Y$ extending a given map $A \to Y$ defined on a subspace $A \subseteq X$ is undecidable, even for $Y$ an even-dimensional sphere. In the present paper,…

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric…

Building on To\"en's work on affine stacks, we develop a certain homotopy theory for schemes, which we call "unipotent homotopy theory." Over a field of characteristic $p>0$, we prove that the unipotent homotopy group schemes…

The Pontryagin-Thom construction provides a fundamental link between cobordism groups and the homotopy groups of Thom spectra. Our main result refines this theorem, providing a more explicit geometric interpretation of the homotopy groups…

We show that the homotopy type of a 4-manifold $M$ whose fundamental group is a finitely presentable $PD_3$-group $\pi$ and with $w_1(M)=w_1(\pi)$ is determined by $\pi$, $\pi_2(M)$, $k_1(M)$ and the equivariant intersection pairing…

This paper studies the homotopy and homeomorphism classifications of $4$-manifolds with boundary. Given $4$-manifolds $X_0$ and $X_1$ with fundamental group $\pi$, we consider the problem of extending a homotopy equivalence $h \colon…

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the…

We study the irreducible components of special loci of curves whose group of symmetries is given as certain group extension. We introduce some relative Hurwitz data, which we show by using mixed \'etale cohomology theory, identifies some…

We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized…

We study the homotopy decompositions of the suspension $\Sigma M$ of an $(n-1)$-connected $(2n+2)$ dimensional Poincar\'{e} duality complex $M$, $n\geq 2$. In particular, we completely determine the homotopy types of $\Sigma M$ of a…

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n >= 2 we supply n homotopic maps that each represent the (n-1)!-th power of a suitable generator of…