Related papers: The homotopy sequence of the algebraic fundamental…

We construct a log algebraic version of the homotopy sequence for a quasi-projective normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.

In this article we prove exactness of the homotopy sequence of overconvergent $p$-adic fundamental groups for a smooth and projective morphism in characteristic $p$. We do so by first proving a corresponding result for rigid analytic…

We show exactness of the homotopy sequence for the logarithmic fundamental group in the case of log smooth, finitely presented, proper and saturated morphisms of fs log schemes over a field. This generalizes earlier results of Hoshi in the…

Unlike Grothendieck's \'etale fundamental group, Nori's fundamental group does not fulfill the homotopy exact sequence in general. We give necessary and sufficient conditions which force exactness of the sequence.

In homotopy theory, exact sequences and spectral sequences consist of groups and pointed sets, linked by actions. We prove that the theory of such exact and spectral sequences can be established in a categorical setting which is based on…

A separable, proper morphism of varieties with geometrically connected fibers induces a homotopy exact sequence relating the \'etale fundamental groups of source, target and fiber. Extending work of dos Santos, we prove the existence of an…

We show that the tame \'etale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically closed fields of characteristic $p \geq 0$.

In this paper we describe the fundamental group-scheme of a proper variety fibered over an abelian variety with rationally connected fibers over an algebraically closed field. We use old and recent results for the Nori fundamental…

Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of 'homotopical exactness', appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be…

The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth…

Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we…

Over any field of characteristic $0$, we prove that the homotopy exact sequence of algebraic fundamental groups for the universal curve with unordered marked points does not split. The same nonsplitting holds for the universal hyperelliptic…

We define the algebraic fundamental group functor of a reductive group scheme over an arbitrary (non-empty) base scheme and prove that this functor is exact.

In this paper, we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications.

In this paper, we investigate the necessary sufficient conditions for the exactness of the homotopy sequence of Nori's fundamental group and apply these to various special situations to regain some classical theorems and give a counter…

We exhibit a map f between aspherical spaces X and Y such that f induces an isomorphism on homotopy groups but, with natural topologies, X and Y fail to have homeomorphic fundamental groups. Thus the topological fundamental group has the…

We define an analog in characteristic $p>0$ of the proalgebraic completion of the topological fundamental group of a complex manifold.

The pro-\'etale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual \'etale fundamental group $\pi_1^{\mathrm{et}}$ defined in SGA1 and the more general group defined in…

Let $\mathbb{F}$ be an algebraically closed field of characteristic zero. In this article we show that isotypic blocks of finite groups are functorially equivalent over $\mathbb{F}$.

We prove that the homotopy type of a map from a Peano continuum into a planar or one-dimensional space is determined by the induced homomorphism of fundamental groups. This provides a new proof that planar sets are aspherical and is used to…