Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

P. Bartłomiejczyk , B. Kamedulski, P. Nowak-Przygodzki – TOPOLOGY AND ITS APPLICATIONS – 2020
We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.

Degree product formula in the case of a finite group action

Let V, W be finite dimensional orthogonal representations of a finite group G. The equivariant degree with values in the Burnside ring of G has been studied extensively by many authors. We present a short proof of the degree product formula for local equivariant maps on V and W.

A Hopf type theorem for equivariant local maps

We study otopy classes of equivariant local maps and prove a Hopf type theorem for such maps in the case of a real finite-dimensional orthogonal representation of a compact Lie group.

The Hopf type theorem for equivariant gradient local maps

We construct a degree-type otopy invariant for equivariant gradient local maps in the case of a real finite-dimensional orthogonal representation of a compact Lie group. We prove that the invariant establishes a bijection between the set of equivariant gradient otopy classes and the direct sum of countably many copies of Z.

The Hopf theorem for gradient local vector fields on manifolds

P. Bartłomiejczyk , P. Nowak-Przygodzki – New York Journal of Mathematics – 2015
We prove the Hopf theorem for gradient local vector fields on manifolds, i.e., we show that there is a natural bijection between the set of gradient otopy classes of gradient local vector fields and the integers if the manifold is connected Riemannian without boundary.

Proper gradient otopies

P. Bartłomiejczyk , P. Nowak-Przygodzki – TOPOLOGY AND ITS APPLICATIONS – 2012
We prove that the inclusion of the space of proper gradient local maps into the space of proper local maps induces a bijection between the sets of the respective otopy classes of these maps.

The homotopy type of the space of gradient vector fields on the two-dimensional disc

P. Bartłomiejczyk , P. Nowak-Przygodzki – GLASGOW MATHEMATICAL JOURNAL – 2012
We prove that the inclusion of the space of gradient vector fields into the space of all vector fields on D^2 non-vanishing in S^1 is a homotopy equivalence

Otopy classes of equivariant maps

W artykule definiuje się stopień topologiczny niezmienniczych odwzorowań lokalnych w przypadku gradientowym i niegradientowym. Wyniki dotyczą relacji pomiędzy tymi dwoma niezmiennikami topologicznymi.