583 these two maps agree up to $m$-th order homotopy.

   583 these two maps agree up to $m$-th order homotopy.

   584 More precisely, one can show that the subcomplex of maps containing the various

   584 More precisely, one can show that the subcomplex of maps containing the various

   585 $e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$.

   585 $e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$.

   586 \nn{now should remark that we have not, in fact, produced a contractible set of maps,

   586 \nn{now should remark that we have not, in fact, produced a contractible set of maps,

   587 but we have come very close}

   587 but we have come very close}

   588 

   589 

   589 

   590 

   590 

   591 

   591 \nn{...}

   592 \nn{...}

   592 

   593