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.
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.
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 Z.
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.