On the digitally quasi comultiplications of digital images

In this article we study the digitally quasi comultiplications of the digital wedge products of pointed digital images. After defining a digitally quasi co-H-space and a digital Whitehead product, we develop a method of how to calculate the cardinal number of digital homotopy classes based on the digitally quasi comultiplications of a pointed digital image as a particular case. We also construct a digitally quasi co-H-space as a digital retract of a given digitally quasi co-H-space.

Algebraic Invariants of Simple 4-Knots

We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L, η, [λ]), where L = Z ⊕ π2(X)⊕π3(X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, η is composition with the Hopf map and [λ] is the orbit of the homotopy class of the longitude in π4(X) under the group of self homotopy equivalences of the universal covering space X′ which induce the identity on L. If K is fibred these invariants determine the fibre, and the natural Z[t,t-1]-module structures on the homotopy groups capture part of the monodromy. Every such triple with L finitely generated as an abelian group (and satisfying the order obviously necessary conditions) may be realized by some fibred simple 4-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.

On distinguishing spaces not homotopy-equivalent

A non-pathological example is given of two topological spaces which have isomorphic homotopy groups, homology groups and cohomology ring and which cannot be distinguished from each other by the Whitehead product structure. A family of examples can be constructed likewise.