Related papers: On genera of polyhedra

We prove an analogue of the Madsen-Weiss theorem for high dimensional manifolds. For example, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S^n x S^n, in the…

Let $M$ denote the moduli space of stable vector bundles of rank $n$ and fixed determinant of degree coprime to $n$ on a non-singular projective curve $X$ of genus $g \geq 2$. Denote by $\cU$ a universal bundle on $X \times M$. We show…

A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and…

Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}^n U_n$ and $S(V_n)$ denote the…

Let $f: X \to Y$ be a regular covering of a surface $Y$ of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$ to…

Let ${\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, smooth curves. For a family $C\to S$ of such curves over a connected base and a geometric point $\xi$ on $S$, the associated monodromy representation is…

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context.…

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. Fix $n\geq 2$, and an integer $d$. A pair $(E,\phi)$ over $X$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $X$ and a section…

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric…

Spectrahedra are affine sections of the cone of positive semidefinite matrices which form a rich class of convex bodies that properly contains that of polyhedra. While the class of polyhedra is closed under linear projections, the class of…

An algebraic variety $X$ is called a homogeneous variety if the automorphism group $\mathrm{Aut}(X)$ acts on $X$ transitively, and a homogeneous space if there exists a transitive action of an algebraic group on $X$. We prove a criterion of…

We consider the category of presheaves of Gamma-spaces, or equivalently, of Gamma-objects in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on…

Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining…

We classify the localizing tensor ideals of the integral stable module category for any finite group $G$. This results in a generic classification of $\mathbb{Z}[G]$-lattices of finite and infinite rank and globalizes the modular case…

Complex structure moduli of a Calabi-Yau threefold in $N=1$ supersymmetric heterotic compactifications can be stabilized by holomorphic vector bundles. The stabilized moduli are determined by a computation of Atiyah class. In this paper, we…

A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular but "fails geometric regularity by a factor of 2"; its combinatorial automorphism group is flag-transitive but its geometric…

We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT).…

In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of…

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the…

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong…