Mathematics

The main objective of this paper is to compute RO(G) -graded cohomology of G -orbits for the group G= C n , where n is a product of distinct primes. We compute these groups for the constant Mackey functor Z – – and for the Burnside ring Mackey functor A – – . Among other things, we show that the groups H – – α G ( S 0 ) are mostly determined by the fixed point dimensions of the virtual representations α , except in the case of A – – coefficients when the fixed point dimensions of α have many zeros. In the case of Z – – coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain G -complexes.

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that their isotropy action is equivariantly formal. In this article we show that ( Z 2 ⊕ Z 2 ) -symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

We investigate the combinatorial data arising from the classification of equivariant homotopy commutativity for cyclic groups of order G= C p 1 ⋯ p n for p i distinct primes. In particular, we will prove a structural result which allows us to enumerate the number of N ∞ -operads for C pqr , verifying a computational result.

In this paper, we study genuine equivariant factorization homology and its interaction with equivariant Thom spectra, which we construct using the language of parametrized higher category theory. We describe the genuine equivariant factorization homology of Thom spectra, and use this description to compute several examples of interest. A key ingredient for our computations is an equivariant nonabelian Poincaré duality theorem, in which we prove that factorization homology with coefficients in a G -space is given by a mapping space. We compute the Real topological Hochschild homology ( THR ) of the Real bordism spectrum M U R and of the equivariant Eilenberg--MacLane spectra H F – – 2 and H Z – – (2) , as well as factorization homology of the sphere S 2σ with coefficients in these Eilenberg--MacLane spectra. In the appendix, Jeremy Hahn and Dylan Wilson compute THR(H Z – – ) .

We develop the theory of equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We construct a good notion of equivariant presheaves, with a suitable sheafification functor. Using these results on equivariant presheaves, we give explicit constructions of products of equivariant sheaves of R-modules. We introduce an equivariant analogue of skyscraper sheaves, which allows us to show that the category of equivariant sheaves of R-modules over a profinite space has enough injectives. This paper also provides the basic theory for results by the authors on giving an algebraic model for rational G-spectra in terms of equivariant sheaves over profinite spaces. For those results, we need a notion of Weyl-G-sheaves over the space of closed subgroups of G. We show that Weyl-G-sheaves of R-modules form an abelian category, with enough injectives, that is a full subcategory of equivariant sheaves of R-modules. Moreover, we show that the inclusion functor has a right adjoint.

We present estimates of number of simplices of given dimension of classical compact Lie groups. As in the previous work \cite{GMP2} the approach is a combination of an estimate of number of vertices with a use of valuation of the covering type by cohomological argument of \cite{GMP} and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the case of exceptional Lie groups we made a complete calculation using the description of their cohomology rings given by the first and third author. For infinite increasing series of Lie groups of growing dimension d the rate of growth of number of simplices of highest dimension is given which extends onto the case of simplices of (fixed) codimension d−i .

We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. The goal of this paper is to present the extension of using the Euler characteristic in higher-dimensional parameter spaces. While topological data analysis of higher-dimensional parameter spaces using stronger invariants such as homology continues to be the subject of intense research, Euler characteristic is more manageable theoretically and computationally, and this analysis can be seen as an important intermediary step in multi-parameter topological data analysis. We show the usefulness of the techniques using artificially generated examples, and a real-world application of detecting diabetic retinopathy in retinal images.

Closed (and simply-connected) manifolds whose dimensions are larger than 4 are classified via sophisticated algebraic and abstract theory such as surgery theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed manifolds in such ways. However, the latter work is, in geometric and constructive ways, not so difficult in a sense. The fact that the dimensions are not high enables us to handle the manifolds via diagrams for example. It is difficult to study higher dimensional manifolds in these ways, although it is natural and important. In the present paper, we present such studies via fold maps, which are higher dimensional versions of Morse functions. The author previously constructed fold maps on 7-dimensional closed and simply-connected manifolds satisfying additional conditions on cohomology rings, including exotic homotopy spheres. This paper presents fold maps on such manifolds of new classes.

We give explicit formulae for differential graded Lie algebra (DGLA) models of 3-cells. In particular, for a cube and an n -faceted banana-shaped 3-cell with two vertices, n edges each joining those two vertices and n bi-gon 2-cells, we construct a model symmetric under the geometric symmetries of the cell fixing two antipodal vertices. The cube model is to be used in forthcoming work for discrete analogues of differential geometry on cubulated manifolds.

A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set X K has n -simplices given by the simplicial maps Δ[n]×K→X . We prove that X K is non-singular whenever X is non-singular. It follows that non-singular simplicial sets form a cartesian closed category with all limits and colimits, but it is not a topos.