# A relaxed evaluation subgroup - Mathematics > Algebraic Topology

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Abstract: Let $f:X\to Y$ be a pointed map between connectedCW-complexes. As a generalization of the evaluation subgroup $G *Y,X;f$, wewill define the {\it relaxed evaluation subgroup} ${\mathcal G} *Y,X;f$ inthe homotopy group $\pi *Y$ of $Y$, which is identified with ${ m Im}\pi *\tilde{ev}$ for the evaluation map $\tilde{ev} :mapX,Y;f\times X\to Y$given by $\tilde{ev} h,x=hx$. Especially we see by using Sullivan model inrational homotopy theory for the rationalized map $f {\Q}$ that ${\mathcalG} *Y {\Q},X {\Q};f {\Q}=\pi *Y\otimes \Q$ if the map $f$ induces aninjection of rational homotopy groups. Also we compare it with more relaxedsubgroups by several rationalized examples.

Autor: Toshihiro Yamaguchi

Fuente: https://arxiv.org/