Index Function and Minimal Cycles

Cycle index sum for non-k-equal configurations

We compute the cycle index sum of the symmetric group action on the homology of the configuration spaces of points in a Euclidean space with the condition that no $k$ of them are equal.

Geometric cycles, index theory and twisted K-homology

We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $α: X \to K(\ZZ, 3)$. We introduce the topological index and the analytical index on the bordism group of $α$-twisted $Spin^c$-manifolds over $(X, α)$, taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topological index and the analytical index. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of $(X, \a)$ analogous to Baum-Douglas's geometric cycl…

Symmetric products of surfaces and the cycle index

We express the signature ${\rm Sign}(SP^m_G(M))$ of the symmetric product $SP^n(M)$ of an (open) surface $M$ in terms of the cycle index $Z(G;\bar x)$ of $G$, a polynomial which originally appeared in P{\' o}lya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfaces $M_{g,k}, M_{g',k'}$ such that the manifolds $SP^{m}(M_{g,k})$ and $SP^{m}(M_{g',k'})$ are often not homeomorphic, although they always have the same homotopy type provided $2g+k = 2g'+k'$ and $k,k'\geq 1$.

Computing Minimal Persistent Cycles: Polynomial and Hard Cases

Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in a purely topological persistence diagram (also termed as barcode). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with $\mathbb{Z}_2$ coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of…

正交丛的各向同性截面的Hopf指数

A Hopf index for isotropic sections of orthogonal bundles

将Hopf指数推广到带有二次型的丛的各向同性截面，得到八个类似物，应用于余截面局部化虚拟循环和DT^4虚拟循环。