2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66188Described the algebraic structure on the space of homotopy classes of cycles with marked topological flags of disks. This space is a non-commutative monoid, with an Abelian quotient corresponding to the group of singular homologies $H_k(M)$. For the marked flag contracted to a point the multiplication becomes commutative and the subgroup of spherical cycles corresponds to the usual homotopy group $π_k(M)$.6 pages LaTeX, 5 figuresAlgebraic TopologyK-Theory and Homologytext