Version 4: Major revision: Sections reordered (Kühnel's question comes earlier). Technical homotopy condition simplified. Added Corollary 8 on odd-dimensional Kühnel's question. Added Conjecture 18 that implies Kühnel's conjecture. Manifolds with boundary treated more carefully. Obstruction treated in the deleted product setting only. Added more details to Table 1
We improve the bound on Kühnel's problem to determine the smallest $n$ such that the $k$-skeleton of an $n$-simplex $Δ_n^{(k)}$ does not embed into a compact PL $2k$-manifold $M$ by showing that if $Δ_n^{(k)}$ embeds into $M$, then $n\leq (2k+1)+(k+1)β_k(M;\mathbb Z_2)$. As a consequence we obtain improved Radon and Helly type results for set systems in such manifolds.
Our main tool is a new description of an obstruction for embeddability of a $k$-complex $K$ into a compact PL $2k$-manifold $M$ via the intersection form on $M$. In our approach we need that for every map $f\colon K\to M$ the restriction to the $(k-1)$-skeleton of $K$ is nullhomotopic. In particular, this condition is satisfied in interesting cases if $K$ is $(k-1)$-connected, for example a $k$-skeleton of $n$-simplex, or if $M$ is $(k-1)$-connected. In addition, if $M$ is $(k-1)$-connected and $k\geq 3$, the obstruction is complete, meaning that a $k$-complex $K$ embeds into $M$ if and only if the obstruction vanishes. For trivial intersection forms, our obstruction coincides with the standard van Kampen obstruction. However, if the form is non-trivial, the obstruction is not linear but rather 'quadratic' in a sense that it vanishes if and only if certain system of quadratic diophantine equations is solvable. This may potentially be useful in attacking algorithmic decidability of embeddability of $k$-complexes into PL $2k$-manifolds.
For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of $S^3$, and some Seifert manifolds. In application, we establish a formula for calculating the linking form of a cyclic branched cover of $S^3$, and develop procedures of computing some Dijkgraaf-Witten invariants.
We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called "thermodynamic" regime. We prove the existence of universal limit laws for the topologies; namely, the random normalized counting measure of connected components (counted according to homotopy type) is shown to converge in probability to a deterministic probability measure. Moreover, we show that the support of the deterministic limiting measure equals the set of all homotopy types for Euclidean geometric complexes of the same dimension as the manifold.
Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $Σ^2M$ as wedge sums of some elementary $\mathbf{A}_3^3$-complexes, which are $2$-connected finite complexes of dimension at most $6$. Furthermore, we utilize the Postnikov square (or equivalently Pontryagin square) to find sufficient conditions for the homotopy decompositions of $Σ^2M$ to desuspend to that of $ΣM$.
Simplex2Vec embeddings for community detection in simplicial complexes
Institute for Scientific Interchange（科学交流研究所）；IMT School for Advanced Studies（IMT高等研究学院）；ICTEAM, Université Catholique de Louvain（鲁汶大学ICTEAM研究所）；Università di Torino（都灵大学）；INFN（意大利国家核物理研究所）；Queen Mary University of London（伦敦玛丽女王大学）；The Alan Turing Institute（艾伦·图灵研究所）；ISI Global Science Foundation（ISI全球科学基金会）
Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the co…
The Geometry of Cochains on Sampled Vietoris-Rips Complexes
University of Edinburgh（爱丁堡大学）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology（系统生物学中心）；Technische Universitat Dresden（德累斯顿工业大学）
