441 \]

   441 \]

   442 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$

   442 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$

   443 which fix $\bd X$.

   443 which fix $\bd X$.

   444 These action maps are required to be associative up to homotopy

   444 These action maps are required to be associative up to homotopy

   445 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that

   445 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that

   447 \nn{repeat diagram here?}

   447 \nn{repeat diagram here?}

   448 \nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}

   448 \nn{restate this with $\Homeo(X\to X')$?  what about boundary fixing property?}

   449 \end{axiom-numbered}

   449 \end{axiom-numbered}

   450 

   450 

   451 We should strengthen the above axiom to apply to families of extended homeomorphisms.

   451 We should strengthen the above axiom to apply to families of extended homeomorphisms.

   965 \]

   965 \]

   966 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$

   966 Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$

   967 which fix $\bd M$.

   967 which fix $\bd M$.

   968 These action maps are required to be associative up to homotopy

   968 These action maps are required to be associative up to homotopy

   969 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that

   969 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that

   971 \nn{repeat diagram here?}

   971 \nn{repeat diagram here?}

   972 \nn{restate this with $\Homeo(M\to M')$?  what about boundary fixing property?}}

   972 \nn{restate this with $\Homeo(M\to M')$?  what about boundary fixing property?}}

   973 

   973 

   974 \medskip

   974 \medskip

   975 

   975