Let $X$ be a $2n$-manifold with a locally standard action of a compact torus $T^n$. If the free part of action is trivial and proper faces of the orbit space $Q$ are acyclic, then there are three types of homology classes in $X$: (1) classes of face submanifolds; (2) $k$-dimensional classes of $Q$ swept by actions of subtori of dimensions $<k$; (3) relative $k$-classes of $Q$ modulo $\partial Q$ swept by actions of subtori of dimensions $\geqslant k$. The submodule of $H_*(X)$ spanned by face classes is an ideal in $H_*(X)$ with respect to the intersection product. It is isomorphic to $(\mathbb{Z}[S_Q]/Θ)/W$, where $\mathbb{Z}[S_Q]$ is the face ring of the Buchsbaum simplicial poset $S_Q$ dual to $Q$; $Θ$ is the linear system of parameters determined by the characteristic function; and $W$ is a certain submodule, lying in the socle of $\mathbb{Z}[S_Q]/Θ$. Intersections of homology classes different from face submanifolds are described in terms of intersections on $Q$ and $T^n$.
Let $X$ be 2n-dimensional compact manifold with a locally standard action of a compact torus. The orbit space $X/T$ is a manifold with corners. Suppose that all proper faces of $X/T$ are acyclic. In the paper we study the homological spectral sequence $E^*_{*,*}\Rightarrow H_*(X)$ corresponding to the filtration of $X$ by orbit types. When the free part of the action is not twisted, we describe the whole spectral sequence in terms of homology and combinatorial structure of $X/T$. In this case we describe the kernel and the cokernel of the natural map $k[X/T]/(l.s.o.p.) \to H_*(X)$, where $k[X/…
We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original graph-theoretical definition could be captured in the usual language for homology theories, so as to make it applicable to more general spaces. In this paper we provide such a general topological framework: we define a homology theory which satisfies the usual axioms, but which maintains the special role for ends that has made this homology work so well for gra…
We show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a homologically locally connected filtration consisting of paracompact Hausdorff spaces is 0.
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.
We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension~1 captures precisely the topological cycle space of graphs but works in any dimension.
We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory. The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenie…
This paper is the last part of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form in arXiv:0912.4213. The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph …
A 2-torus manifold is a closed smooth manifold of dimension $n$ with an effective action of a 2-torus group $(\Z_2)^n$ of rank $n$, and it is said to be locally standard if it is locally isomorphic to a faithful representation of $(\Z_2)^n$ on $\R^n$. This paper studies the equivariant classification of locally standard 2-torus manifolds.
Let $X$ be a higher rank symmetric space or a Bruhat-Tits building of dimension at least $2$ such that the isometry group of $X$ has property $(T)$. We prove that for every torsion free lattice $Γ\subset {\rm Isom} X$ any homology class in $H_1(Γ\backslash X,\mathbb F_2)$ has a representative cycle of total length $o_X({\rm Vol}(Γ\backslash X))$. As an application we show that $\dim_{\mathbb F_2} H_1(Γ\backslash X,\mathbb F_2)=o_X({\rm Vol}(Γ\backslash X)).$
For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus our computation gives the homology and cohomology of Bestvina-Brady groups. We compute the cohomological dimension of each of these groups over any field and over any subring of the rationals.
The {\it torus manifolds} have been defined and studied by M. Masuda and T. Panov (arXiv:math.AT/0306100) who in particular describe its cohomology ring structure. In this note we shall describe the topological $K$-ring of a class of torus manifolds (those for which the orbit space under the action of the compact torus is a {\it homology polytope} whose {\it nerve} is a {shellable} simplicial complex) in terms of generators and relations. Since these torus manifolds include the class of quasi-toric manifolds this is a generalisation of earlier results due to the author and P. Sankaran (arXiv: …
We consider the orbit type filtration on a manifold $X$ with locally standard action of a compact torus and the corresponding homological spectral sequence $(E_X)^r_{*,*}$. If all proper faces of the orbit space $Q=X/T$ are acyclic, and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms are equal to the $h'$-numbers of the Buchsbaum simplicial poset $S_Q$ dual to $Q$. Betti numbers of $X$ depend only on the orbit space $Q$ but not on the characteristic function. If $X$ is a slightly different object, namely the model space $X=(P…
We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Mardešić and Prasolov showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author which establishes the consistency, relative to a weakly compact cardinal, of $\mathrm{lim}^s \mathbf{A} = 0$ for all $s \geq …
We introduce an elementary way of constructing principal (Z_2)^m-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard (Z_2)^m-actions on closed manifolds for all m>0, and then give a general way to construct all such (Z_2)^m-actions from the orbit space. Some related topology problems are also studied.
We give a necessary and sufficient condition for the orientability of a locally standard 2-torus manifold with a fixed point which generalizes previous results of Nakayama-Nishimura in 2005 and Soprunova-Sottile in 2013. We construct manifolds with boundary where the boundary is a disjoint union of locally standard 2-torus manifolds. We discuss equivariant oriented cobordism class of locally standard 2-torus manifolds.
We consider the category of all locally Lipschitz contractible metric spaces and all locally Lipschitz maps, which is a wide class of metric spaces, including all finite dimensional Alexandrov spaces and all CAT spaces. We also consider the chain complex of normal currents with compact support in a metric space in the sense of Ambrosio and Kirchheim. In the present paper, its homology is proved to be a homotopy invariant on the category. To prove this result, we define a new topology on a space of Lipschitz maps between arbitrary metric spaces. This topology is proved to coincide with the usua…
We show that a set with an action of a locally finite-dimensional free partially commutative monoid and the corresponding semicubical set have isomorpic homology groups. We build a complex of finite length for the computing homology groups of any asynchronous transition system with finite maximal number of mutually independent events. We give examples of computing the homology groups.
