Ursula Gritsch
We prove that the space of gauge equivalence classes of U(1)-invariant connections on some SU(2)-principle bundles over the 4-sphere S^4 is weakly homotopy equivalent to a component of the second loop space of the 2-sphere S^2.
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We show that $S^n \vee S^m$ is $\mathbb{Z}/p^r$-hyperbolic for all primes $p$ and all $r \in \mathbb{Z}^+$, provided $n,m \geq 2$, and consequently that various spaces containing $S^n \vee S^m$ as a $p$-local retract are $\mathbb{Z}/p^r$-hyperbolic. We then give a $K$-theory criterion for a suspension $ΣX$ to be $p$-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian $ΣGr_{k,n}$ is $p$-hyperbolic for all odd primes $p$ when $n \geq 3$ and $0<k<n$. We obtain similar results for some related spaces.
We study the asymptotic behavior of the homotopy groups of simply connected finite $p$-local complexes, and define a space to be locally hyperbolic if its homotopy groups have exponential growth. Under some certain conditions related to the functorial decomposition of loop suspension, we prove that the suspended finite complexes are locally hyperbolic if suitable but accessible information of the homotopy groups is known. In particular, we prove that Moore spaces are locally hyperbolic, and other candidates are also given.
In this paper, we show that for every non-nilpotent hyperbolic map $f$ on an infra-nilmanifold, the $\textrm{HPer}(f)$ is cofinite in $\mathbb{N}$. This generalizes a similar result for expanding maps. Moreover, we prove that for every nilpotent map $f$ on an infra-nilmanifold, $\textrm{HPer}(f)=\{1\}$.
Homotopy exponents of polyhedral products
本文研究了棱柱积的同伦指数与Moore猜想，证明了在特定条件下，棱柱积在任何奇素数处均无同伦指数，并给出了判断棱柱积在有理范畴下超几何的准则。
In this paper we study the relationship between the homology and homotopy of a space at infinity and at its boundary. Firstly, we prove that if a locally connected, connected, $δ$-hyperbolic space that is acted upon geometrically by a group has trivial homotopy at infinity then the first Čech homotopy group is trivial. Secondly, we prove that if a hyperbolic group on a finite field has trivial $i^{th}$ homology at infinity then the boundary of the group has trivial $i^{th}$ Steenrod homology. This result turns out to be important in addressing an open problem related to Cannon's conjecture.
The monopole h-invariants from a topological perspective
本文从拓扑视角研究 3-流形的单极子 h-不变量，考察其对系数选择的依赖性，并补充了若干专家熟知但文献中难以查证的性质证明。
Let A be a geometric arrangement such that codim(x) > 1 for every x in A. We prove that, if the complement space M(A) is rationally hyperbolic, then there exists an injective from a free Lie algebra L(u,v) to the homotopy Lie algebra of M(A).
We study the notion of $p$-quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every $p$-quasihomotopy class of Newtonian maps contains a minimizer of the $p$-energy if the target has hyperbolic fundamental group.
We prove that the Teichmüller space $\mathcal{T}^{<0}(M)$ of negatively curved metrics on a hyperbolic manifold $M$ has nontrivial $i$-th rational homotopy groups for some $i> \dim M$. Moreover, some elements of infinite order in $π_i B\mbox{Diff}(M)$ can be represented by bundles over $S^i$ with fiberwise negatively curved metrics.
This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful. This study provides a Künneth formula for twisted coarse cohomology. We obtain the Gromov boundary of a hyperbolic proper geodesic metric space as a quotient of its Higson corona.
Exotic structures on hyperbolic manifolds via the EO-theory of Projective Spaces
本文利用复射影空间的EO-理论，在多种维度构造了与给定双曲流形同伦等价但非微分同胚的负曲率流形，并证明了奇异球面上自由光滑S^1和S^3作用的存在性。
Some asymptotic formulae for torsion in homotopy groups
受有理同伦群秩渐近估计及局部双曲性研究启发，本文证明了同伦群挠秩的两个渐近公式，并据此得到了多种空间在悬垂后同伦群挠秩的显式定量渐近下界。
Let A be a subspace arrangement with a geometric lattice such that codim(x) > 1 for every x in A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum of the orthogonal subspaces is a direct sum. The homotopy type of M(A) is also given: it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincare duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that M(A) is rationally elliptic, and sho…
We show that the homotopy groups of a Moore space $P^n(p^r)$, where $p^r \neq 2$, are $\mathbb{Z}/p^s$-hyperbolic for $s \leq r$. Combined with work of Huang-Wu, Neisendorfer, and Theriault, this completely resolves the question of when such a Moore space is $\mathbb{Z}/p^s$-hyperbolic for $p \geq 5$, or when $p=2$ and $r \geq 6$. We also give a criterion in ordinary homology for a space to be $\mathbb{Z}/p^r$-hyperbolic, and deduce some examples.
Homotopy of Simply Connected Complexes with a Spherical Pair
研究了具有球对的连通复形中循环空间分解及其对同调理论的影响，揭示了不同惰性层次之间的差异及高维中同伦等价的多样性。
We study the rational homotopy of the moduli space ${\mathcal N}_X$ of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface $X$ of genus $g\geq 2$. The symplectic group $Aut(H_1(X,{\mathbb Z}))=Sp(2g,{\mathbb Z})$ has a natural action on the rational homotopy groups $π_n({\mathcal N}_X) \otimes {\mathbb Q}$. We prove that this action extends to an action of $Sp(2g,{\mathbb C})$ on $π_n({\mathcal N}_X) \otimes {\mathbb C}$. We also show that $π_n({\mathcal N}_X) \otimes {\mathbb C}$ is a non-trivial $Sp(2g,{\mathbb C})$-representation fo…
Local hyperbolicity, inert maps and Moore's conjecture
该研究证明了在特定条件下，流形的底空间是局部双曲的，并在足够大的素数处满足摩尔猜想。
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivial in many cases. In complex dimension 9, we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.
This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate' the components via the companionship trees associated to the knot. The knots with the simplest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy-type of these components of K were computed by Hatcher. In the case the companionship tree has height, we give a fibre-bundle description of those components of K, recursively, in…
Compact aspherical solenoids
本文研究由 CW 复形与覆盖映射逆极限获得的紧致非球面螺线管，利用形状理论对映射进行同伦分类，并证明了自同伦等价的 Dehn-Nielsen 型定理，推广了 Odden 的结果。
It is proven that the homotopy time-slice axiom for many types of algebraic quantum field theories (AQFTs) taking values in chain complexes can be strictified. This includes the cases of Haag-Kastler-type AQFTs on a fixed globally hyperbolic Lorentzian manifold (with or without time-like boundary), locally covariant conformal AQFTs in two spacetime dimensions, locally covariant AQFTs in one spacetime dimension, and the relative Cauchy evolution. The strictification theorems established in this paper prove that, under suitable hypotheses that hold true for the examples listed above, there exist…
Polar decompositions of complex varieties and tropical homotopy theory
本文提出了一种复形种的角对pants分解方法，并证明其与本质投影超平面补集的同伦等价性，结合热带几何技术进行具体应用。
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