On homotopy nilpotency of loop spaces of Moore spaces

Abstract

\n\t <jats:p>Let <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink= http://www.w3.org/1999/xlink mime-subtype= png xlink:href= S000843952100028X_inline1.png />\n\t\t<jats:tex-math>\n$M(A,n)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> be the Moore space of type <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink= http://www.w3.org/1999/xlink mime-subtype= png xlink:href= S000843952100028X_inline2.png />\n\t\t<jats:tex-math>\n$(A,n)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> for an Abelian group <jats:italic>A</jats:italic> and <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink= http://www.w3.org/1999/xlink mime-subtype= png xlink:href= S000843952100028X_inline3.png />\n\t\t<jats:tex-math>\n$n\\ge 2$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>. We show that the loop space <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink= http://www.w3.org/1999/xlink mime-subtype= png xlink:href= S000843952100028X_inline4.png />\n\t\t<jats:tex-math>\n$\\Omega (M(A,n))$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> is homotopy nilpotent if and only if <jats:italic>A</jats:italic> is a subgroup of the additive group <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink= http://www.w3.org/1999/xlink mime-subtype= png xlink:href= S000843952100028X_inline5.png />\n\t\t<jats:tex-math>\n$\\mathbb {Q}$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> of the field of rationals. Homotopy nilpotency of loop spaces <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink= http://www.w3.org/1999/xlink mime-subtype= png xlink:href= S000843952100028X_inline6.png />\n\t\t<jats:tex-math>\n$\\Omega (M(A,1))$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> is discussed as well.</jats:p>