Degrees of Self-Maps of Products

Neofytidis, Christoforos

doi:10.1093/imrn/rnw227

“…The proof is based on Thom's representation of homology classes by manifolds and straightforward calculations in singular homology and cohomology; similar arguments appear in related work on domination of/by product manifolds [6,7,12].…”

Section: Products Of Strongly Inflexible Manifoldsmentioning

confidence: 95%

Exotic Finite Functorial Semi-Norms on Singular Homology

Fauser¹,

Löh²

2018

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“…Recently, many results about mapping degree via simplicial volume and bounded cohomology have arised in the literature [LK09], [LS09], [BBI13], [Neo17], [Neo18], [FM19], [DLSW19]. Among these applications, we are primarily interested in both the work by Bucher, Burger and Iozzi [BBI13] and the one by Derbez, Liu, Sun and Wang [DLSW19].…”

Section: Maximal Cocycles and Local Isometriesmentioning

confidence: 99%

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Moraschini

¹

,

Savini

²

2020

Transformation Groups

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Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

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“…The interest in the relation between the mapping degree of continuous maps and the volume of manifolds led to a rich and fruitful literature [35,38,39,27]. Derbez et al [18,Proposition 3.1] were able to express the volume of the pullback of a representation ρ along a continuous map f as the product of the mapping degree of f with the volume of ρ.…”

Section: Natural Maps For Zimmer's Cocyclesmentioning

confidence: 99%

Natural Maps for Measurable Cocycles of Compact Hyperbolic Manifolds

Savini

¹

2021

J. Inst. Math. Jussieu

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Let $\operatorname {\mathrm {{\rm G}}}(n)$ be equal to either $\operatorname {\mathrm {{\rm PO}}}(n,1),\operatorname {\mathrm {{\rm PU}}}(n,1)$ or $\operatorname {\mathrm {\textrm {PSp}}}(n,1)$ and let $\Gamma \leq \operatorname {\mathrm {{\rm G}}}(n)$ be a uniform lattice. Denote by $\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ the hyperbolic space associated to $\operatorname {\mathrm {{\rm G}}}(n)$ , where $\operatorname {\mathrm {{\rm K}}}$ is a division algebra over the reals of dimension d. Assume $d(n-1) \geq 2$ . In this article we generalise natural maps to measurable cocycles. Given a standard Borel probability $\Gamma $ -space $(X,\mu _X)$ , we assume that a measurable cocycle $\sigma :\Gamma \times X \rightarrow \operatorname {\mathrm {{\rm G}}}(m)$ admits an essentially unique boundary map $\phi :\partial _\infty \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \partial _\infty \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $\phi _x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are atomless for almost every $x \in X$ . Then there exists a $\sigma $ -equivariant measurable map $F: \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $F_x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are differentiable for almost every $x \in X$ and such that $\operatorname {\mathrm {\textrm {Jac}}}_a F_x \leq 1$ for every $a \in \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ and almost every $x \in X$ . This allows us to define the natural volume $\operatorname {\mathrm {\textrm {NV}}}(\sigma )$ of the cocycle $\sigma $ . This number satisfies the inequality $\operatorname {\mathrm {\textrm {NV}}}(\sigma ) \leq \operatorname {\mathrm {\textrm {Vol}}}(\Gamma \backslash \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}})$ . Additionally, the equality holds if and only if $\sigma $ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \operatorname {\mathrm {{\rm G}}}(n) \leq \operatorname {\mathrm {{\rm G}}}(m)$ , modulo possibly a compact subgroup of $\operatorname {\mathrm {{\rm G}}}(m)$ when $m>n$ . Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.

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Degrees of Self-Maps of Products

Cited by 12 publications

References 19 publications

Exotic Finite Functorial Semi-Norms on Singular Homology

Exotic Finite Functorial Semi-Norms on Singular Homology

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Natural Maps for Measurable Cocycles of Compact Hyperbolic Manifolds

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