José Carlos Corrêa Eidam, Paolo Piccione
In this article, we show that the Fredholm Lagrangian Grassmannian is homotopy equivalent with the space of compact perturbations of a fixed lagrangian. As a corollary, we obtain that the Maslov index with respect to a lagrangian is a isomorphism between the fundamental group of the Fredholm Lagrangian Grassmannian and the integers.
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The homotopy classification of proper Fredholm maps of index one
本研究在前期已完成非正指标真Fredholm映射同伦分类的基础上，引入新的数值不变量，完成了单连通希尔伯特流形到其模型空间的指标为1的真Fredholm映射的同伦类计算。
We point out a flaw in papers by Daniel Biss devoted to homotopy type of matroid Grassmannians.
Non-linear proper Fredholm maps and the stable homotopy groups of spheres
本文利用球面稳定同伦群分类 Hilbert 空间间的非线性真 Fredholm 映射，给出满射及其非平凡核，并证明有向情形与稳定同伦群一一对应。
Rational stable homotopy type of equivariant projective spaces and Grassmannians
本文研究了等效复射影空间和格拉斯曼流形在复表示下的有理稳定性类型，通过显式分裂展示了其结构特征。
We classify the homotopy classes of proper Fredholm maps from an infinite dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli-Smale mod 2 degree and the absolute value of the oriented degree.
Spaces of unbounded Fredholm operators: I. Homotopy equivalences
本文研究图拓扑下无界Fredholm算子空间及其子空间，证明它们与经典有界算子空间的自然映射是同伦等价，揭示无界本质正Fredholm算子子空间表示奇K理论，且各空间的可逆算子子空间可缩。
The incidence correspondence in the grassmannian which determines the tautological bundle defines a map between cycle spaces on grassmannians. These cycle spaces decompose canonically into a product of Eilenberg-MacLane spaces. These decompositions and the associated maps are calculated up to homotopy.
We study the analytic and homotopy properties of some infinite dimensional Grassmannians, useful for developing a Morse theory for infinite dimensional manifolds. We study the space of Fredholm pairs of a Hilbert space, we determine its homotopy type, and we define a determinant bundle over it. We study the space of compact perturbations of a given closed linear subspace, and the related concept of essential Grassmannian.
We describe the fundamental group and second homotopy group of ordered $k-$point sets in $Gr(k,n)$ generating a subspace of fixed dimension.
Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity
研究从射影直线到拉格朗日格拉斯曼流形的基于代数映射空间，通过箭图描述实现代数几何精细化，得到同构，其拓扑实现恢复经典同伦等价。
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。
Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian
本文针对复简约群的Ran Grassmannian的$\u211e_2$-代数结构民间结论，通过纯代数方法证明开球包含诱导Beilinson–Drinfeld Grassmannian同伦等价，得到等变分层同痕的更强同伦不变性结果。
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.
Endomorphisms of Equivariant Algebraic $K$-theory
本文证明了有限常值群概形作用下等变代数K-理论由Grassmannian的余极限表示，并通过显式计算Grassmannian的等变K-理论，证明了等变动机空间与无限Grassmannian的自同态集一致。
On The Acyclic MacPhersonian
本文研究非循环MacPhersonian $\text{AM}_n^d$，在球形分离系偏序集上证明其与Grassmann流形 $G(d,n-1)$ 的同伦等价性，分析平均曲率流策略并结合定理得出相关结论，提出开放问题。
We prove an h-principle with boundary condition for a certain class of topological spaces valued sheaves. The techniques used in the proof come from the study of the homotopy type of the cobordism categories, and they are of simplicial and categorical nature. Applying the main result of this paper to a certain sheaf we find another proof of the Madsen-Weiss theorem, that describes the homotopy type of the classifying space of the cobordism category as the loop space of the Thom space of the complement of the tautological bundle over the Grassmannians.
Schubert cells of mixed type in complex Lagrangian Grassmannians
本文描述了复 Lagrange Grassmann 流形的 CW 分解，显式计算了粘贴映射的度，并由此得出了实与复 Lagrange Grassmann 流形的同伦扩张性质及部分挠类的可缩性。
The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold, an algebraic variety, a vector bundle, a classifying space, among many more structures; furthermore; it affords an analogue of Schubert calculus and its (co)homology and homotopy groups may be…
In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we construct a nullhomotopy whose \emph{thickness} (Lipschitz constant in the space variable) is $C(m,n)(L+1)$ and whose \emph{width} (Lipschitz constant in the time variable) is $C(m,n)(L+1)^2$. More generally, we prove a similar result for maps $f:X \to Y$ for any compact Riemanni…
We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural $Spin^c$-Dirac operators. As an application we give new examples of non-trivial Hamiltonian fibrations, that have not been previously detected by other methods. As one crucial ingredient we construct a potentially new homotopy equivalence map, with a certain naturality property, from $BU$ to the space of index $0$ Fredholm operators on a Hilbert space, using el…
Our main theorem is that the inclusion of a Birkhoff variety in the affine Grassmannian is a homotopy equivalence. We also construct analogues of tubular neighborhoods for Birkhoff and Schubert varieties. We include some observations on torus-equivariant cohomology.
Simplicial approximation to CW complexes with spherical Delaunay triangulations
本文提出基于球面德劳内三角化的细分方法，改进了传统单纯形近似在计算中的不足，通过将星条件分解为几何和组合问题，显著减少顶点数量，成功构造出与Grassmannian和Stiefel流形同调等价的单纯复形。
We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classe…
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