松弛评价子群
A relaxed evaluation subgroup
摘要：
设$f:X\to Y$是连通CW-复形之间的点映射。作为评价子群$G_*(Y,X;f)$的推广，我们将在$Y$的同伦群$π_*(Y)$中定义{ \it松弛评价子群} ${\mathcal G}_*(Y,X;f)$，它与$\tilde{ev} (h,x)=h(x)$给出的评价映射$\tilde{ev} :map(X,Y;f)\times X\to Y$的${\rm Im} π_*(\tilde{ev})$相同。特别是对于有理化映射$f_{\Q}$，利用有理同伦理论中的Sullivan模型，我们看到如果映射$f$诱导有理同伦群的注入，则${\mathcal G}_*(Y_{\Q},X_{\Q};f_{\Q})=π_*(Y)\otimes \Q$.通过几个有理化的例子，我们还将它与更松弛的子群进行了比较。
英文摘要：
Let $f:X\to Y$ be a pointed map between connected CW-complexes. As a generalization of the evaluation subgroup $G_*(Y,X;f)$, we will define the {\it relaxed evaluation subgroup} ${\mathcal G}_*(Y,X;f)$ in the homotopy group $π_*(Y)$ of $Y$, which is identified with ${\rm Im} π_*(\tilde{ev})$ for the evaluation map $\tilde{ev} :map(X,Y;f)\times X\to Y$ given by $\tilde{ev} (h,x)=h(x)$. Especially we see by using Sullivan model in rational homotopy theory for the rationalized map $f_{\Q}$ that ${\mathcal G}_*(Y_{\Q},X_{\Q};f_{\Q})=π_*(Y)\otimes \Q$ if the map $f$ induces an injection of rational homotopy groups. Also we compare it with more relaxed subgroups by several rationalized examples.
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