Maps to spaces in the genus of infinite quaternionic projective space

某些流形上的四元数射影丛全空间的双商模型

The biquotient model of the total space of the projectivization quaternionic bundles over certain manifolds

本文计算四元数流形上四元数切丛诱导的射影丛全空间的有理同伦型，并给出底空间为四元数射影空间时的双商模型表示。

拟射影四元数空间的子空间{L}usternik-{S}chnirelmann范畴

Subspace {L}usternik-{S}chnirelmann category of quasi-projective quaternionic spaces

本文利用四元数对数证明辛群Sp(n)的拟射影子空间Q_n的子空间Lusternik-Schnirelmann范畴为2，推广了n=2的已知结果，并与cat Q_3=3形成对比。

论悬浮四元数射影平面的同伦群及其应用

On the Homotopy Groups of the Suspended Quaternionic Projective Plane and Applications

本文计算了悬浮四元数射影平面同伦群在特定范围内的2、3分量，并应用于局部化悬浮四元数射影空间的CW复形分类及自smash积分解。

Torus action on quaternionic projective plane and related spaces

For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated fixed points the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $\mathbb{H}P^2/T^3\cong S^5$ and $S^6/T^2\cong S^4$, for the homogeneous spaces $\mathbb{H}P^2=Sp(3)/(Sp(2)\times Sp(1))$ and $S^6=G_2/SU(3)$. Here the maximal tori of the corresponding Lie groups $Sp(3)$ and $G_2$ act on the homogeneous spaces by the left multiplication. Next we consider the quaternio…

On the symmetric squares of complex and quaternionic projective space

The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a solution for the complex and quaternionic projective spaces $KP^n$, by taking advantage of their rich geometrical structure. Our description is in terms of generators and relations, and our methods entail ideas that have appeared in the literature of quantum chemistry, theoretical physics, and combinatorics. We deal first with the case $KP^\infty$, and proceed by …

A topological obstruction to existence of quaternionic Plucker map

We prove the following statement motivated by the quaternionic linear algebra: there is no continuous map from the quaternionic Grassmannian to the quaternionic projective space such that the pull back of the first Pontryagin class of the tautological bundle over the projective space is equal to the first Pontryagin class of the tautological bundle over the Grassmannian.

四元射影平面连通和上实向量丛的复结构与稳定复结构

Complex and Stable Complex Structures on Real Vector Bundles over Connected Sums of Quaternionic Projective Planes

本文通过庞特里亚金类刻画四元射影平面连通和上实向量丛的（稳定）复结构，纠正了1974年Sato-Suzuki关于殆复结构的错误断言，给出精确充要条件。

四元射影空间连通和上实向量丛的稳定复结构

Stable Complex Structures On Real Vector Bundles Over Connected Sums Of Quaternionic Projective Spaces

本文通过分析复化映射的像，用庞特里亚金类刻画了四元射影空间连通和上实向量丛的稳定复结构，证明了仅当 n=2 且 ℓ-m 为偶数时该空间稳定殆复，并彻底否定了 n≥3 时殆复结构的存在性。

四元数射影平面的一个15顶点三角剖分

A 15-vertex triangulation of the quaternionic projective plane

本研究针对1992年Brehm与Kühnel构造的15顶点8维单纯复形是否为四元数射影平面最小三角剖分的悬而未决问题，通过实现Gaifullin的算法计算其第一庞特里亚金类，最终证实该复形确为四元数射影平面的最小三角剖分。

射影空间连通和的配边结构集

Concordance structure set of connected sum of projective spaces

该研究计算了8到16维范围内复射影空间与四元数射影空间连通和的配边结构集，证明了连通和的配边惯性群为各配边惯性群之和，并对比了流形与其连通和的配边结构集。

On trivialities of Chern classes

A finite $CW$-complex $X$ is $C$-trivial if for every complex vector bundle $ξ$ over $X$, the total Chern class $c(ξ)=1$. In this note we completely determine when each of the following spaces are $C$-trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces and suspensions of stunted quaternionic projective spaces.

Dualization invariance and a new complex elliptic genus

We define a new elliptic genus psi on the complex bordism ring. With coefficients in Z[1/2], we prove that it induces an isomorphism of the complex bordism ring modulo the ideal which is generated by all differences P(E)-P(E*) of projective bundles and their duals onto a polynomial ring on 4 generators in degrees 2, 4, 6 and 8. As an alternative geometric description of psi, we prove that it is the universal genus which is multiplicative in projective bundles over Calabi-Yau 3-folds. With the help of the q-expansion of modular forms we will see that for a complex manifold M, the value psi(M) i…

The $\hat{A}$ genus as a projective volume form on the derived loop space

In the present work, we extend our previous work with Gwilliam by realizing $\hat{A}(X)$ as the projective volume form associated to the BV operator in our quanitization of a one-dimensional sigma model. We also discuss the associated integration/expectation map. We work in the formalism of $L_\infty$ spaces, objects of which are computationally convenient presentations for derived stacks. Both smooth and complex geometry embed into $L_\infty$ spaces and we specialize our results in both of these cases.

Algebraic cycles and the classical groups II: Quaternionic cycles

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel-Whitney classes. In this sequel, we establish corresponding results in the case where V has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a simi…