413 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
413 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
414 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
414 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
415 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
415 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
416 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
416 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
417 |
417 |
418 \begin{thm} \label{thm:CH} |
418 \begin{thm} \label{thm:CH} \label{thm:evaluation}% |
419 For $n$-manifolds $X$ and $Y$ there is a chain map |
419 For $n$-manifolds $X$ and $Y$ there is a chain map |
420 \eq{ |
420 \eq{ |
421 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
421 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
422 } |
422 } |
423 well-defined up to homotopy, |
423 well-defined up to homotopy, |
424 such that |
424 such that |
425 \begin{enumerate} |
425 \begin{enumerate} |
426 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
426 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
427 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
427 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and |
428 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
428 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
429 the following diagram commutes up to homotopy |
429 the following diagram commutes up to homotopy |
430 \begin{equation*} |
430 \begin{equation*} |
431 \xymatrix@C+2cm{ |
431 \xymatrix@C+2cm{ |
432 CH_*(X, Y) \otimes \bc_*(X) |
432 CH_*(X, Y) \otimes \bc_*(X) |