902 The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. |
902 The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. |
903 |
903 |
904 We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. |
904 We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. |
905 Define $h$ to be zero on bidegree $(0, k-1)$. |
905 Define $h$ to be zero on bidegree $(0, k-1)$. |
906 Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$. |
906 Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$. |
907 Using Lemma \ref{extension_lemma}, construct a homotopy $q$ from $p$ to $p'$ which is adapted to $\cU$. |
907 Using Lemma \ref{extension_lemma}, construct a homotopy (rel $\bd$) $q$ from $p$ to $p'$ which is adapted to $\cU$. |
908 Define $h$ at $p\otimes b$ to be $q\otimes b$. |
908 Define $h$ at $p\otimes b$ to be $q\otimes b$. |
909 Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. |
909 Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. |
910 Let $a$ be that portion of $\bd(p'\otimes b')$ which intersects the boundary of |
910 Let $s$ denote the sum of the $q$'s from the previous step for generators |
911 bidegree $(1, k-2)$ stuff. |
911 adjacent to $(\bd p')\otimes b'$. |
912 Apply Lemma \ref{extension_lemma} to $p'$ plus the diffeo part of $h(a)$ |
912 \nn{need to say more here} |
913 (rel the outer boundary of said part), |
913 Apply Lemma \ref{extension_lemma} to $p'+s$ |
914 yielding a family of diffeos $q'$. |
914 yielding a family of diffeos $q'$. |
915 \nn{definitely need to say this better} |
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916 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. |
915 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. |
917 Continuing in this way, we define all of $h$. |
916 Continuing in this way, we define all of $h$. |
918 |
917 |
919 The homotopy $r$ is constructed similarly. |
918 The homotopy $r$ is constructed similarly. |
920 |
919 |