Narrow Results

In this paper, we study notions of persistent homotopy groups of compact metric spaces. We pay particular attention to the case of fundamental groups, for which we obtain a more precise description via a persistent version of the notion of discrete fundamental groups due to Berestovskii–Plaut and Barcelo et al. Under fairly mild assumptions on the spaces, we prove that the persistent fundamental group admits a tree structure which encodes more information than its persistent homology counterpart. We also consider the rationalization of the persistent homotopy groups and by invoking results of Adamaszek–Adams and Serre, we completely characterize them in the case of the circle. Finally, we establish that persistent homotopy groups enjoy stability in the Gromov–Hausdorff sense. We then discuss several implications of this result including that the critical spectrum of Plaut et al. is also stable under this notion of distance.

Given a simply connected, closed 4-manifold, we prove that the homotopy groups of such a manifold are determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth 4-manifold with second Betti number at least three the number of geometrically distinct periodic geodesics of length at most l grows exponentially as a function of l.

We determine, in terms of and , the homotopy groups of certain groups of invertibles and of certain equivalence classes in the infinite Grassmann space on a Hilbert C*--module. These results provide various interpretations of .

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group π₃. This generalizes a result of Gutierrez–Ratcliffe who relate the intersection of two subgroups with the computation of π₂. Let K be a two-dimensional CW-complex with subcomplexes K₁, K₂, K₃ such that K = K₁ ∪ K₂ ∪ K₃ and K₁ ∩ K₂ ∩ K₃ is the 1-skeleton K¹ of K. We construct a natural homomorphism of π₁(K)-modules

In 1947, in the paper “Theory of Braids,” Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk $B_n$, denoted by ${\widehat{B}}_n$. In this paper, we extend Goldsmith’s answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion of homotopy generalized string links on surfaces, which form a group which is a proper quotient of the braid group on a surface $B_n(M)$, denoting it by ${\widehat{B}}_n(M)$. We then give a presentation of the group ${\widehat{B}}_n(M)$ and find that the Goldsmith presentation is a particular case of our main result, when we consider the surface $M$ to be the disk. We close with a brief discussion surrounding the importance of having such a fixed construction available in the literature.

Let $M$ be a subset of vector space or projective space. The authors define generalized configuration space of $M$ which is formed by $n$-tuples of elements of $M$, where any $k$ elements of each $n$-tuple are linearly independent. The generalized configuration space gives a generalization of Fadell’s classical configuration space, and Stiefel manifold. Denote generalized configuration space of $M$ by $W_{k,n}(M)$.

For studying topological property of the generalized configuration spaces, the authors calculate homotopy groups for some special cases. This paper gives the fundamental groups of generalized configuration spaces of $ℝP^m$ for some special cases, and the connections between the homotopy groups of generalized configuration spaces of $S^m$ and the homotopy groups of Stiefel manifolds. It is also proved that the higher homotopy groups of generalized configuration spaces $W_{k,n}(S^m)$ and $W_{k,n}(ℝP^m)$ are isomorphic.

Let BP be the Brown-Peterson spectrum at an odd prime p, and L₂ denote the Bousfield localization functor with respect to . The Ravenel spectrum T(1) is characterized by BP_*(T(1)) = BP_*[t₁] on the primitive generator t₁. In this paper, we determine the homotopy groups π_*(L₂M ∧ T(1)) for the mod p Moore spectrum M.

In a previous paper, the authors show some examples of compact symplectic solvmanifolds, of dimension six, which are cohomologically Kähler and they do not admit Kähler metrics because their fundamental groups cannot be the fundamental group of any compact Kähler manifold. Here we generalize such manifolds to higher dimension and, by using Auroux symplectic submanifolds [3], we construct four-dimensional symplectically aspherical manifolds with nontrivial π₂ and with no Kähler metrics.