This paper compares two invariants of foliated manifolds which seem to measure the non-Hausdorffness of the leaf space: the transversal length on the fundamental group and the foliated Gromov norm on the homology. We consider foliations with the property that the set of singular simplices transverse to the foliation satisfies a weakened version of the Kan extension property. (We prove that this assumption is fairly general: it holds for all fibration-covered foliations, in particular for all foliations of 3-manifolds without Reeb components.) For such foliations we show that vanishing of the transversal length implies triviality of the foliated Gromov norm, and, more generally, that uniform bounds on the transversal length imply explicit bounds for the foliated Gromov norm.
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth. Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coinc…
A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classe…
Let $X^{n}$ be an arbitrary oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597-607) we have constructed a map $t:\mathcal{N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^+)$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal{N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important nontrivial question arose…
Functorial semi-norms are semi-normed refinements of functors such as singular (co)homology. We investigate how different types of representability affect the (non-)triviality of finite functorial semi-norms on certain functors or classes. In particular, we consider representable functors, generalised cohomology theories, and so-called weakly flexible homology classes in singular homology and l1-homology.
We describe a construction of the cyclotomic structure on topological Hochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are able to define versions of topological cyclic homology ($TC$) and TR-theory relative to a cyclotomic commutative ring spectrum $A$. We describe spectral sequences computing this relative theory $_ATR$ in terms of $TR$ over the sphere spectrum and vice versa. Furthermore, our construction …
We show that the Connes-Consani semi-norm on singular homology with real coefficients, defined via s-modules, coincides with the ordinary $\ell^1$-semi-norm on singular homology in all dimensions.
In the paper the Čech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of Čech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-Čech compactifications of metrizable spaces.
We define the homology of a simplicial set with coefficients in a Segal's $Γ$-set ($\mathbf S$-module). We show the relevance of this new homology with values in $\mathbf S$-modules by proving that taking as coefficients the $\mathbf S$-modules at the archimedean place over the structure sheaf on $\overline{Spec\mathbb Z}$ introduced in our previous work, one obtains on the singular homology with real coefficients of a topological space $X$, a norm equivalent to the Gromov norm. Moreover, we prove that the two norms agree when $X$ is an oriented compact Riemann surface.
Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of $3$-dimensional hyperbolic manifolds that converges to $\mathbb{H}^3$ in the Benjamini--Schramm topology while its normalized torsion in the first homology is dense in $[0,\infty]$. In dimension $d\geq 4$ a somewhat precise estimate is given for the number of …
In this paper we study $\mathcal M(X)$, the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of $X$, via a map $Ψ$ from $\mathcal M(X)$ into the quotient of $K(X)=[X,BSO]$ by the action of the group of homotopy classes of simple self equivalences of $X$. The map $Ψ$ describes which bundles over $X$ can occur as normal bundles of manifolds in $\mathcal M(X)$. We determine the image of $Ψ$ when $X$ belongs to a certain class of homology spheres. In particular, we find conditions on elements of $K(X)$ that guarantee they are pullbacks of normal bundles of manifolds …
We generalize the results of a previous paper of ours to compact Lie groups. Using a recently developed ordinary equivariant homology and cohomology, we define equivariant Poincare complexes with the properties that (1) every compact G-manifold is an equivariant Poincare complex, (2) every finite equivariant Poincare complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the Pi-Pi Theorem holds for equivariant Poincare pairs under suitable gap hypotheses. The nice behavior of the ordinary equivariant homology and cohomology theories allows …
Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of "connection" as a new kind of degeneracy. In this paper, for a cubical set with connections, we show that the connections generate an acyclic subcomplex of the chain complex of the cubical set. In particular, our results show that the homology groups of a cubical set with connections are independent of whether we normalize by the connections or we do not, that is, connections do not contribute to any nontrivial cycle in the homology groups of the cubical set.
Let $N \subset M$ be a submanifold embedding of spin manifolds of some codimension $k \geq 1$. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that $M$ does not admit a metric of positive scalar curvature if $k = 2$ and the Dirac operator of $N$ has non-trivial index, provided that suitable conditions are satisfied. In the cases $k=1$ and $k=2$, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer $\mathrm{KO}_\ast(\mathrm{C}^{\ast} π_1 M)\to \mathrm{KO}_{\ast - k}(\mathrm{C}^\ast π_1 N)$ which maps the index class of…
We extend the work of Robinson and Turner to use hypothesis testing with persistence homology to test for measurable differences in shape between point clouds from three or more groups. Using samples of point clouds from three distinct groups, we conduct a large-scale simulation study to validate our proposed extension. We consider various combinations of groups, samples sizes and measurement errors in the simulation study, providing for each combination the percentage of $p$-values below an alpha-level of 0.05. Additionally, we apply our method to a Cardiotocography data set and find statisti…
