On topological homotopy groups and relation to Hawaiian groups

Abstract

By generalizing the whisker topology on the nth homotopy group of pointed space (X, x 0), denoted by π wh n (X, x 0), we show that π wh n (X, x 0) is a topological group if n ≥ 2. Also, we present some necessary and sufficient conditions for π wh n (X, x 0) to be discrete, Hausdorff and indiscrete. Then we prove that L n (X, x 0) the natural epimorphic image of the Hawaiian group H n (X, x 0) is equal to the set of all classes of convergent sequences to the identity in π wh n (X, x 0). As a consequence, we show that L n (X, x 0) ∼ = L n (Y, y 0) if π wh n (X, x 0) ∼ = π wh n (Y, y 0), but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally n-simply connected spaces and n-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of n-loop space coincide. Finally, we show that n-SLT paths can transfer π wh n and hence L n isomorphically along its points.

On Hawaiian homology groups

“…In the following theorem, we present an equivalent condition for paths to transfer Hawaiian groups isomorphically. The condition n-SLT introduced in [4], is sufficient but not necessary. Recall that a path γ from x 0 to x 1 in a space X is called a small n-loop transfer (abbreviated to n-SLT), if for every open neighborhood U of x 0 , there exists an open neighborhood V of x 1 , such that for every n-loop β :…”