Mathematics

We prove that every action of A 5 on a finite 2 -dimensional contractible complex has a fixed point.

In this series of two articles, we prove that every action of a finite group G on a finite and contractible 2 -complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the acyclic 2 -dimensional G -complexes constructed by Oliver and Segev. In the first part we develop the necessary theory and cover the cases where G= PSL 2 ( 2 n ) , G= PSL 2 (q) with q??(mod8) or G=Sz( 2 n ) . The cases G= PSL 2 (q) with q??(mod8) are addressed in the second part.

In this second part we prove that, if G is one of the groups PSL 2 (q) with q>5 and q??(mod24) or q??3(mod24) , then the fundamental group of every acyclic 2 -dimensional, fixed point free and finite G -complex admits a nontrivial representation in a unitary group U(m) . This completes the proof of the following result: every action of a finite group on a finite and contractible 2 -complex has a fixed point.

Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacians (PLs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the non-harmonic spectra of PLs give rise to additional geometric analysis of the shape of the data. In this work, we develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES), to enable broad applications of PSGs in science, engineering, and technology. To ensure the reliability and robustness of HERMES, we have validated the software with simple geometric shapes and complex datasets from three-dimensional (3D) protein structures. We found that the smallest non-zero eigenvalues are very sensitive to data abnormality.

We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group Γ with additional cohomological properties. For Γ= Z 2 we recover the conjugation spaces of Hausmann, Holm and Puppe. For Γ=U(1) we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials.

We introduce a distance function between simplicial complexes and study several of its properties.

For r≥1 , the r -independence complex of a graph G is a simplicial complex whose faces are subset I⊆V(G) such that each component of the induced subgraph G[I] has at most r vertices. In this article, we determine the homotopy type of r -independence complexes of certain families of graphs including complete s -partite graphs, fully whiskered graphs, cycle graphs and perfect m -ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.

We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the E 2 -topological Hochschild cohomology of certain Thom spectra (denoted A , B , and T(n) ) related to Ravenel's X( p n ) . We show that these conjectures imply that the orientations MSpin→ko and MString→tmf admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs H F p as a Thom spectrum, to construct BP⟨n−1⟩ , ko , and tmf as Thom spectra (albeit over T(n) , A , and B respectively, and not over the sphere). This interpretation of BP⟨n−1⟩ , ko , and tmf offers a new perspective on Wood equivalences of the form bo∧Cη≃bu : they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of BP⟨n−1⟩ also provides a different lens on the nilpotence theorem. Finally, we prove a C 2 -equivariant analogue of our construction, describing HZ – – – – as a Thom spectrum.

To a compact Lie group G one can associate a space E(2,G) akin to the poset of cosets of abelian subgroups of a discrete group. The space E(2,G) was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that G is abelian if and only if π i (E(2,G))=0 for i=1,2,4 . This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply--connected if and only if the group is abelian.

In this survey, we discuss two research areas related to Massey's higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley-Reisner rings of simplicial complexes.