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