Piotr Bartłomiejczyk

The exponential law for partial, local and proper maps and its application to otopy theory

We introduce the topology in the set of local maps and prove the exponential law for partial, local and proper maps.

Spectral sequences and detailed connection matrices

We introduce detailed connection matrices. We prove that the spectral sequence can be reconstructed from a detailed connection matrix in the category of filtered differential vector spaces.

The Conley index and spectral sequences

We define spectral sequences associated with Morse decompositions of a compact metric space. We prove the existence and uniqueness of such spectral sequences for continuous dynamical systems.

On the space of equivariant local maps

We introduce the space of equivariant local maps and present the full proof of the splitting theorem for the set of otopy classes of such maps in the case of a representation of a compact Lie group.

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

We prove that for n > 1 the space of proper maps P0(n, k) and the space of local maps F0(n, k) are not homotopy equivalent.

On the topology of spaces of partial and local maps

Abstract. We prove the exponential law for partial and local maps.

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

We present a short proof that if two gradient maps on the two-dimensional disc have the same degree, then they are gradient homotopic.