Related papers: Rationalized Evaluation Subgroups of a Map and the…

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

Traditionally, homotopy groups in $G$-equivariant stable homotopy theory have been graded over $\text{RO}(G)$, the real representation ring of $G$. It is arguably more natural to grade homotopical structures over the Picard group of the…

Let $\Gamma$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $\Gamma$ has a finite index subgroup $G$ which has a rational growth series with respect to a natural generating set. We do this by…

The purpose of this paper is to give some solutions for the classification problem in fibration theory by using the homotopy sequences of fibrations (sequences of $n$-th homotopy groups $ \pi_{n}(S,s_{o}) $ of total spaces of fibrations).…

In this paper we investigate the $directed$ $normalizing$ $graph$ associated with a group $G$, defined as the simple directed graph whose vertices are the elements of $G$, with an arrow from $x$ to $y$ whenever the subgroup $\langle x…

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…

We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In…

Let k be an algebraically closed field of characteristic 0, and let f be a morphism of smooth projective varieties from X to Y over the ring k((t)) of formal Laurent series. We prove that if a general geometric fiber of f is rationally…

Rationale discovery is defined as finding a subset of the input data that maximally supports the prediction of downstream tasks. In the context of graph machine learning, graph rationale is defined to locate the critical subgraph in the…

Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has a rationalization $Y \to Y_{(0)}$ which is well-understood. Early on it was found that the induced map $[X,Y] \to [X,Y_{(0)}]$ on sets of mapping classes is…

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain estimates of the (co)homological dimension of groups G…

We calculate the higher homotopy groups of the Deligne-Getzler infinity-groupoid associated to a nilpotent L-infinity algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

Let $f : X \rightarrow Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between \'etale fundamental groups $f_* :…

We provide a simple condition on rational cohomology for the total space of a pullback fibration over a connected sum to have the rational homotopy type of a connected sum, after looping. This takes inspiration from recent work of Jeffrey…

Let $X$ be a closed Riemann surface. When $X$ is embedded into a projective space, the first rational cohomology group can be concretely obtained from the monodromy in the family of its smooth hyperplane sections by C. Schnell's tube…

In this paper, we investigate some topics around the closed image $S$ of a rational map $\lambda$ given by some homogeneous elements $f_1,...,f_n$ of the same degree in a graded algebra $A$. We first compute the degree of this closed image…

For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M--> N.We show that the evaluation map from the space of maps to the manifold N induces…

We generalize theorems of McGibbon-Roitberg, Iriye, and Meier on the relations between phantom maps and rational homotopy, and apply them to provide new calculational examples of the homotopy sets Ph(X, Y ) of phantom maps and the subsets…

Given two real algebraic varieties X and Y, we denote by R(X,Y) the set of all regular maps from X to Y. The set R(X,Y) is regarded as a topological subspace of the space C(X,Y) of all continuous maps from X to Y endowed with the…

This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup $(G,+)$ of rationals, two main questions are addressed: first, what are the model-theoretic…