Magnitude and magnitude homology of filtered sets enriched categories

In this article, we give a framework for studying the Euler characteristic and its categorification of objects across several areas of geometry, topology and combinatorics. That is, the magnitude theory of filtered sets enriched categories. It is a unification of the Euler characteristic of finite categories and it the magnitude of metric spaces, both of which are introduced by Leinster. Our definitions cover a class of metric spaces which is broader than the original ones, so that magnitude (co)weighting of infinite metric spaces can be considered. We give examples of the magnitude from various research areas containing the Poincar\'{e} polynomial of ranked posets and the growth function of finitely generated groups. In particular, the magnitude homology gives categorifications of them. We also discuss homotopy invariance of the magnitude homology and its variants. Such a homotopy includes digraph homotopy and r-closeness of Lipschitz maps. As a benefit of our categorical view point, we generalize the notion of Grothendieck fibrations of small categories to our enriched categories, whose restriction to metric spaces is a notion called metric fibration that is initially introduced by Leinster. It is remarkable that the magnitude of such a fibration is a product of those of the fiber and the base. We especially study fibrations of graphs, and give examples of graphs with the same magnitude but are not isomorphic.Comment: 35 pages, 1 figur

Small Covers, infra-solvmanifolds and curvature

It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover being homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structure of the underlying simple polytopes.Comment: 22 pages, no figur

On the split structure of lifted groups

Let ▫$wp colon tilde{X} to X$▫ be a regular covering projection of connected graphs with the group of covering transformations ▫$rm{CT}_wp$▫ being abelian. Assuming that a group of automorphisms ▫$G le rm{Aut} X$▫ lifts along $wp$ to a group ▫$tilde{G} le rm{Aut} tilde{X}$▫, the problem whether the corresponding exact sequence ▫$rm{id} to rm{CT}_wp to tilde{G} to G to rm{id}$▫ splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither ▫$tilde{G}$▫ nor the action ▫$Gto rm{Aut} rm{CT}_wp$▫ nor a 2-cocycle ▫$G times G to rm{CT}_wp$▫, are given. Explicitly constructing the cover ▫$tilde{X}$▫ together with ▫$rm{CT}_wp$▫ and ▫$tilde{G}$▫ as permutation groups on ▫$tilde{X}$▫ is time and space consuming whenever ▫$rm{CT}_wp$▫ is largethus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group)one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever ▫$rm{CT}_wp$▫ is elementary abelian.Naj bo ▫$wp colon tilde{X} to X$▫ regularna krovna projekcija povezanih grafov, grupa krovnih transformacij ▫$rm{CT}_wp$▫ pa naj bo abelova. Ob predpostavki, da se grupa avtomorfizmov ▫$G le rm{Aut} X$▫ dvigne vzdolž ▫$wp$▫ do grupe ▫$tilde{G} le rm{Aut} tilde{X}$▫, podrobno analiziramo problem, ali se ustrezno eksaktno zaporedje ▫$rm{id} to rm{CT}_wp to tilde{G} to G to rm{id}$▫ razcepi glede na Cayleyevo dodelitev napetosti, ki rekonstruira projekcijo do ekvivalence natančno. V gornjem kombinatoričnem sestavu je razširitev podana samo implicitno: podani niso ne ▫$tilde{G}$▫ ne delovanje ▫$Gto rm{Aut} rm{CT}_wp$▫ ne 2-kocikel ▫$G times G to rm{CT}_wp$▫. Eksplicitno konstruiranje krova ▫$tilde{X}$▫ ter ▫$rm{CT}_wp$▫ in ▫$tilde{G}$▫ kot permutacijskih grup na ▫$tilde{X}$▫ je časovno in prostorsko zahtevno vselej, kadar je ▫$rm{CT}_wp$▫ veliktako je uporaba implementiranih algoritmov (na primer, HasComplement v Magmi) vse prej kot optimalna. Namesto tega pokažemo, da lahko najnujnejšo informacijo o delovanju in 2-kociklu učinkovito izluščimo neposredno iz napetosti (ne da bi eksplicitno konstruirali krov in dvignjeno grupo)zdaj bi bilo mogoče uporabiti standardno metodo reduciranja problema na reševanje sistema linearnih enačb nad celimi števili. Vendar tukaj uberemo malce drugačen pristop, ki sploh ne zahteva nobenega poznavanja kohomologije. Časovno in prostorsko zahtevnost formalno analiziramo za vse primere, ko je ▫$rm{CT}_wp$▫ elementarna abelova

Computation of First Cohomology Groups of Finite Covers

AbstractWe give several applications of standard methods of group cohomology to some problems arising in model theory concerning finite covers. We prove a conjecture of the author that forG-finite, ℵ0-categorical structures the kernels of minimal superlinked finite covers have bounded rank. We show that the cohomology groups associated to finite covers of certain structures (amongst them, the primitive, countable, totally categorical structures) have to be finite. From this we deduce that the finite covers of these structures are determined up to finitely many possibilities by their kernels

Investigating Abstract Algebra Students' Representational Fluency and Example-Based Intuitions

The quotient group concept is a difficult for many students getting started in abstract algebra (Dubinsky et al., 1994; Melhuish, Lew, Hicks, and Kandasamy, 2020). The first study in this thesis explores an undergraduate, a first-year graduate, and second-year graduate students' representational fluency as they work on a "collapsing structure", quotient, task across multiple registers: Cayley tables, group presentations, Cayley digraphs to Schreier coset digraphs, and formal-symbolic mappings. The second study characterizes the (partial) make-up of two graduate learners' example-based intuitions related to orbit-stabilizer relationships induced by group actions. The (partial) make-up of a learner's intuition as a quantifiable object was defined in this thesis as a point viewed in R17, 12 variable values collected with a new prototype instrument, The Non-Creative versus Creative Forms of Intuition Survey (NCCFIS), 2 values for confidence in truth value, and 3 additional variables: error to non-error type, unique versus common, and network thinking. The revised Fuzzy C-Means Clustering Algorithm (FCM) by Bezdek et al. (1981) was used to classify the (partial) make-up of learners' reported intuitions into fuzzy sets based on attribute similarity