298 Next we show that $g$ induces a surjection on homology.

   299 Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union 

   300 of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$.

   301 \nn{maybe should refer to elsewhere in this paper where we made a very similar argument}

   302 Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$.

   303 It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$

   304 retracts onto $S_*$.

   305 

   306 Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$.

   307 We claim that $S_{\le k}$ lies in the image of $g$.

   308 Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$,

   309 $j \le k$.

   310 We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls

   311 in $M$.

   312 It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is 

   313 disjoint from the $n{-}1$-skeleton of $K$.

   314 It is now easy to see that $c$ is in the image of $g$.

   315 

   316 Next we show that $g$ is injective on homology.

   317 

   318 

   319 

   320