arXiv subjects
Publications and source records attributed to Piotr Nowak-Przygodzki.
We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ and proper gradient maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ with the Brouwer degree greater or equal to zero.
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.
We prove that the inclusion of the space of gradient local maps into the space of all local maps from Hilbert space to itself induces a bijection between the sets of the respective otopy classes of these maps, where by a local map we mean a compact perturbation of identity with a compact preimage of zero.
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 $W$.
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 $\mathbb{Z}$.