Related papers: Rigid rational homotopy types

In this talk, I report on three theorems concerning algebraic varieties over a field of characteristic $p>0$. a) over a finite field of cardinal $q$, two proper smooth varieties which are geometrically birational have the same number of…

We study the rational homotopy types of classifying spaces of automorphism groups of smooth simply connected manifolds of dimension at least five. We give dg Lie algebra models for the homotopy automorphisms and the block diffeomorphisms of…

We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same…

We describe a collection of higher homotopy operations which determine the rational homotopy type of a simply-connected space X. These are described in terms of simplicial resolutions of successive approximations (L^k,\alpha} to the Quillen…

The homotopy type of the complement of a complex coordinate subspace arrangement is studied by fathoming out the connection between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy…

The intended model of the homotopy type theories used in Univalent Foundations is the infinity-category of homotopy types, also known as infinity-groupoids. The problem of higher structures is that of constructing the homotopy types needed…

We show that if a complex has free finitely generated reduced homology groups for two consecutive dimensions and trivial homology for all other dimensions, then it must have the homotopy type of a wedge of spheres of two consecutive…

Given a closed $n$-manifold, we consider the set of simple homotopy types of $n$-manifolds within its homotopy type, called its simple homotopy manifold set. We characterise it in terms of algebraic K-theory, the surgery obstruction map,…

We investigate a new case of rigidity in stable homotopy theory which is the rigidity of the $K(1)$-local stable homotopy category $\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ at $p=2$. In other words, we show that recovering higher homotopy…

We prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.

Let $k$ be a perfect field of characteristic $p > 0$, $W_n = W_n(k)$. For separated $k$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with…

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer,…

Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…

This short note is extracted from Section 3 and Appendix of the paper entitled with Intersection de Rham complexes in positive characteristic by the same named authors, where an explicit infinite homotopy from a Higgs complex to the…

We classify smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid in the sense that any subvariety of S having the same homology class as S_0 is induced by the action of…

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes…

We compare the log motivic stable homotopy category and the usual motivic stable homotopy category over a perfect field admitting resolution of singularities. As a consequence, we show that the log motivic stable homotopy groups are…

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…

This is the first in a series of papers in which we construct and study a new $p$-adic cohomology theory for varieties over Laurent series fields $k(\!(t)\!)$ in characteristic $p$. This will be a version of rigid cohomology, taking values…

Given a suitable functor T:C -> D between model categories, we define a long exact sequence relating the homotopy groups of any X in C with those of TX, and use this to describe an obstruction theory for lifting an object G in D to C.…