Gradient otopies of gradient local maps

Path components of the space of gradient vector fields on the two-dimensional disc

“…In [1] we prove Parusiński's theorem for gradient otopy classes. Although the results presented here and in [1] are partially similar, they are formally independent.…”

“…In [1] we prove Parusiński's theorem for gradient otopy classes. Although the results presented here and in [1] are partially similar, they are formally independent. Furthermore, the otopy relation is more flexible, since it relates maps with not necessarily the same domain (so called local maps), but it is also weaker.…”

“…Furthermore, the otopy relation is more flexible, since it relates maps with not necessarily the same domain (so called local maps), but it is also weaker. As a consequence, the methods used here differ essentially from those in [1].…”

On the topology of spaces of partial and local maps

“…the domain of may vary with . However, not the whole space of partial maps but its subspaces consisting of local and proper maps have turned out to be more useful in applications (see [2][3][4]7]). Most of these applications are based on studying di erent otopy classes of local and proper maps, because for such maps otopy is a natural counterpart of homotopy (otopy relates maps with not necessarily the same domain).…”

On the homotopy equivalence of the spaces of proper and local maps

“…The notion of local maps is introduced in [6] and, independently, in [9]. The relation between gradient and usual local maps (also in the equivariant case) is studied in [2][3][4]. Finally, in [5] authors introduce the topology on the set of local maps and prove that the inclusion of the space of proper maps into the space of local maps is a weak homotopy equivalence if we restrict ourselves to local maps with domains in R + and ranges in R .…”