Uniform cohomological expansion of uniformly quasiregular mappings

Abstract

Let f : M → M be a uniformly quasiregular self-map of a compact, connected, and oriented Riemannian n-manifold M without boundary, n 2. We show that, for k ∈ {0, . . . , n}, the induced homomorphism f * :is the kth singular cohomology of M , is complex diagonalizable and the eigenvalues of f * have absolute value (deg f ) k/n . As an application, we obtain a degree restriction for uniformly quasiregular self-maps of closed manifolds. In the proof of the main theorem, we use a Sobolev-de Rham cohomology based on conformally invariant differential forms and an induced push-forward operator.

Fibers of monotone maps of finite distortion

“…There are several variations of conformal cohomology theories in use: see e.g. [11], [14], and [25]. Since we generally do not assume higher integrability from our maps, the best suited one for our current application is the one from [25].…”

“…[11], [14], and [25]. Since we generally do not assume higher integrability from our maps, the best suited one for our current application is the one from [25]. It is the cohomology of the chain complex W d CE,loc (∧ * M ) given by…”

“…Hence, we refrain from repeating the argument. Note that for the part of the result involving H k CE (M ), a highly detailed explanation of the proof has been given in [25,Section 4]; the compactly supported version has however not been stated previously to our knowledge. Theorem 3.4.…”

Entropy in uniformly quasiregular dynamics

“…Let f : M → M be a uniformly quasiregular self-map of degree at least 2 on a closed, connected, and oriented Riemannian n-manifold M which is not a rational cohomology sphere. Then h(f ) = log deg f. It follows from [15] that s(f * ) = deg f for non-constant uniformly quasiregular self-maps f : M → M . Theorem 1.1 therefore yields the equality…”

Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

“…A BLD-map between locally geodesic, oriented cohomology n-manifolds induces a natural pull-back operator on currents, which commutes with the boundary. In constructing the pull-back of currents, we use the duality of metric currents and polylipschitz forms, developed in [25], and develop a push-forward operator for polylipschitz forms under BLD-maps, analogous to a push-forward of differential forms on manifolds under quasiregular mappings discussed in [16].…”