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 classi cations are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from R^n to R^n and proper gradient maps from R^2 to R^2 with the Brouwer degree greater or equal to zero.

We show that there exist two proper gradient vector fields on Rn which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

The main theorem of this paper states that Morse cohomology groups in a Hilbert space are isomorphic to the cohomological Conley index. It is also shown that calculating the cohomological Conley index does not require finite-dimensional approximations of the vector field. Further directions are discussed.

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.

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.