equal
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80 \end{lemma} |
80 \end{lemma} |
81 |
81 |
82 \begin{proof} |
82 \begin{proof} |
83 Since both complexes are free, it suffices to show that the inclusion induces |
83 Since both complexes are free, it suffices to show that the inclusion induces |
84 an isomorphism of homotopy groups. |
84 an isomorphism of homotopy groups. |
85 To show that it suffices to show that for any finitely generated |
85 To show this it in turn suffices to show that for any finitely generated |
86 pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
86 pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
87 \[ |
87 \[ |
88 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
88 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
89 \] |
89 \] |
90 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
90 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |