相关论文: On the homology of the Harmonic Archipelago

We study three different topologies on the moduli space $\mathscr{H}^{\rm loc}_m$ of equivariant isometry classes of $m$-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally…

A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for…

Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using…

We prove uniqueness of equivariant harmonic maps into irreducible symmetric spaces of non-compact type and Euclidean buildings associated to isometric actions by Zariski dense subgroups.

We compute the \v{C}ech homotopy groups of the $m$-dimensional infinite earring space $\mathbb{E}_m$, i.e. a shrinking wedge of $m$-spheres. In particular, for all $n,m\geq 2$, we prove that $\check{\pi}_n(\mathbb{E}_m)$ is isomorphic to a…

This article shows several new methods for proofs on Kan complexes while using them to give a compact introduction to the homotopy groups of these complexes. Then more advanced objects are studied starting with homology and the Hurewicz…

In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We…

We develop techniques for studying fundamental groups and integral singular homology of symmetric Delta-complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points.…

We extend some basic results from the singular homology theory of topological spaces to the setting of \v{C}ech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We…

In this paper we study the relationship between the homology and homotopy of a space at infinity and at its boundary. Firstly, we prove that if a locally connected, connected, $\delta$-hyperbolic space that is acted upon geometrically by a…

We compute the homology of the space of equivariant loops on the classifying space of a simplicial monoid $M$ with anti-involution, provided $\pi_0 (M)$ is central in the homology ring of $M$. The proof is similar to McDuff and Segal's…

We extend the former results of Ganea and the two of the authors with Takeda on the homotopy commutativity of the loop spaces of Hermitian symmetric spaces such that the loop spaces of all irreducible symmetric spaces but $\mathbb{C}P^3$…

We calculate rational homology groups of spaces of non-resultant (i.e. having no non-trivial common zeros) systems of homogemeous quadratic polynomials in R^3

The resultant veriety in the space of systems of homogeneous polynomials of given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of…

We exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also analyze the…

We compute the homology of random \v{C}ech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erd\H{o}s-R\'enyi phase…

Let K be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of GL_{n+1}(K)$and the space of harmonic cochains defined on the Bruhat-Tits building of GL_{n+1}(K), the…

In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group is…

We prove a homotopy invariance result for the first cohomology group of the special unitary group $\mathrm{SU}_3(F[t])$ with coefficients in irreducible representations of $\mathrm{PGL}_2(F)$. The main theorem establishes that this…

We set up and solve the initial value problem for equivariant harmonic maps of cohomogeneity one manifolds, i.e. we show the local existence of a harmonic map in the neighborhood of a singular orbit. Furthermore, we present some theory of…