n维克莱因瓶的实Bott流形结构及其有理Betti数

n-dimensional Klein bottles

An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

带气泡的假克莱因瓶

A fake Klein bottle with bubble

本文针对拓扑学中是否存在与带气泡克莱因瓶同基本群、同欧拉示性数但不同伦的有限2-复形的存在性问题给出了解答。

Topological Complexity of the Klein bottle

We show that the (normalized) topological complexity of the Klein bottle is $4$. We also show that, for any $g\geq 2$, $TC(N_g)=4$. This completes the recent work by Dranishnikov on the topological complexity of non-orientable surfaces.

An approach to the topological complexity of the Klein bottle

Recently, Cohen and Vandembroucq proved that the reduced topological complexity of the Klein bottle is 4. Simultaneously and independently, we announced a proof of the same result. Mistakes were found in our argument, which was quite different than theirs. After correcting these, we found that our description of the obstruction class agreed with theirs. Our approach to showing that this obstruction is nonzero failed to do so, while theirs did not fail. Here we discuss our approach, which deals more directly with the simplicial structure of the Klein bottle.

Moduli spaces of vector bundles over a real curve: Z/2-Betti numbers

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obtained by Liu and Schaffhauser.

多重分次Betti数与Hilbert函数的稳定性

On the stability of multigraded Betti numbers and Hilbert functions

本文针对多参数持续同调中的多重分次Betti数，引入高效可计算的瓶颈型相异度函数，证明其关于交错距离的稳定性，并给出2参数情形下Hilbert函数的1-Wasserstein稳定性结果。