693 lines
23 KiB
TeX
693 lines
23 KiB
TeX
\chapter{The Eilenberg-Zilber module}
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The functions of this module are related to the chain complex of the
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pro\-duct of two simplicial sets.
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They implement the Eilenberg-Mac Lane homomorphism (function {\tt eml})
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and the Alexander-Whitney homomorphism (function {\tt aw}).
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\vskip 0.45cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt eml} {\em ssx ssy} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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build\index{homomorphism!Eilenberg-Mac Lane} the Eilenberg-Mac Lane homomorphism $\cal EML$, also called
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the {\em shuffle homomorphism}\footnote{{\bf Marvin Greenberg}, {\it Lectures on Algebraic Topology},
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W.A. Benjamin Inc, 1967.}.
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The arguments {\em ssx} and {\em ssy} represent some simplicial sets $X$ and $Y$.
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The homomorphism
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$${\cal EML}:{\cal C}_*(X) \otimes {\cal C}_*(Y) \longrightarrow {\cal C}_*(X \times Y)$$
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is defined by:
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$${\cal EML}(\alpha \otimes \beta)=
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\sum{{\varepsilon(\varpi)}(\eta_{j_q}\ldots \eta_{j_1}\alpha,\, \eta_{i_p}\ldots \eta_{i_1}\beta)},$$
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where $X\times Y$ is the simplicial cartesian product of the two simplicial sets $X$ and $Y$, $\alpha$ (resp. $\beta$)
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is a geometric simplex of $X$ (resp. $Y$), $\eta_k$ is the $k^{th}$ degeneracy operator, the sum is
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over all permutations $\varpi= (i_1 \ldots i_p j_1 \ldots j_q)$ of $(0 \ldots p+q-1)$ such that
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$$ i_1 < i_2 < \cdots < i_p,\quad j_1 < j_2 < \cdots < j_q, \quad p+q=n$$
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and $\varepsilon(\varpi)$ is the signature of the corresponding permutation. The sum at the right hand part determines
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a decomposition of the geometric product $\alpha \times \beta$ into a combination of generators of the
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simplicial cartesian product (see the simplicial sets chapter). The combination takes in account the orientation of every
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simplex. These generators are non-degenerate simplices of the cartesian product but their projections
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are in general degenerate simplices of $X$ and $Y$ (see the examples). \par}
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{\leftskip=5mm
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{\tt aw} {\em ssx ssy} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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build\index{homomorphism!Alexander-Whitney} the Alexander-Whitney chain homomorphism.
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The arguments {\em ssx} and {\em ssy} represent some simplicial sets $X$ and $Y$.
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The homomorphism
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$${\cal AW}:{\cal C}_*(X \times Y) \longrightarrow {\cal C}_*(X) \otimes {\cal C}_*(Y)$$
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is defined by the following rule: let $\sigma$ be an $n$-simplex and let us define
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two operators, $\lambda_p$ and $\rho_q$:
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$$\lambda_p: {\cal C}_n(X) \longrightarrow {\cal C}_{n-p}(X),\quad (n \geq p),$$
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$$\rho_q: {\cal C}_n(Y) \longrightarrow {\cal C}_{n-q}(X),\quad (n \geq q),$$
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respectively by
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$$\lambda_p(\sigma)=\partial_{p+1}\ldots \partial_n \sigma$$
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and
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$$\rho_q(\sigma)=\partial_0\ldots\partial_{n-q} \sigma.$$
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Now, for the generator $(\alpha,\beta)$ of degree $n$ in ${\cal C}_*(X \times Y)$, the Alexander-Whitney homomorphism
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is defined by:
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$${\cal AW}(\alpha, \beta)=\sum_{p=0}^n{\lambda_p(\alpha) \otimes \rho_{n-p}(\beta)}.$$
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This sum can be interpreted as a decomposition, up to a homotopy, of the simplex $\alpha \times \beta$
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($\alpha$ {\bf and} $\beta$ can be degenerate even if $\alpha \times \beta$ is not) in terms of (tensor)
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products of non-degenerate simplices.
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\par}
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{\leftskip=5mm
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{\tt phi} {\em ssx ssy} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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build the homotopy chain morphism
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$$\Phi: {\cal C}_*(X \times Y) \longrightarrow {\cal C}_*(X \times Y)$$
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satisfying the formula
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$$d \circ \Phi + \Phi \circ d = Id - {\cal EML} \circ {\cal AW}.$$
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\par}
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}
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\newpage
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{\parindent=0mm
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{\leftskip=5mm
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{\tt ez} {\em ssx ssy} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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build the Eilenberg--Zilber reduction as shown by the following dia\-gram, where $X={\mathit ssx}$
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and $Y={\mathit ssy}$.
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$$
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\diagram{
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{{\cal C}_*(X \times Y)} & \stackrel{\Phi}{\longrightarrow} & {}^s{{\cal C}_*(X \times Y)} \cr
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{\scriptstyle {\cal AW}} \downarrow \uparrow {\scriptstyle {\cal EML}} \cr
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{{\cal C}_*(X) \otimes {\cal C}_*(Y)} \cr
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}
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$$
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\par}}
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\begin{center}
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-------------
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\end{center}
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\newpage
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\subsection* {Examples}
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%
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\vskip 0.40cm
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\centerline{\includegraphics{eml1.eps}}
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\vskip 0.40cm
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%
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As simplicial sets $X$ and $Y$, let us use two copies of $\Delta^{\N}$, that we may build
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by the lisp statement {\tt (soft-delta-infinity)} (Though we repeat this statement
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in the following line, there will be only a simplicial set created). We have chosen the {\em soft}
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version, to inspect the results more easily.
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{\footnotesize\begin{verbatim}
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(setf eml-mrp (eml (soft-delta-infinity)(soft-delta-infinity))) ==>
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[K11 Morphism (degree 0)]
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\end{verbatim}}
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Let us apply the Eilenberg-Mac Lane homomorphism to the tensor product
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$(0\, 1)\otimes (2\, 3)$:
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{\footnotesize\begin{verbatim}
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(? eml-mrp 2 (tnpr 1 (d(dlop-ext-int '(0 1)))
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1 (d(dlop-ext-int '(2 3))))) ==>
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----------------------------------------------------------------------{CMBN 2}
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<-1 * <CrPr 0 0-1 1 2-3>>
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<1 * <CrPr 1 0-1 0 2-3>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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\newpage
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The decomposition is illustrated by the previous geometric diagram, where for instance
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a degenerate simplex like $\eta_1(0\, 1)$ is noted $(0\, 1\, 1)$.
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\par
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For the tensor product $(0\,1\,2)\otimes (3\,4)$, an intuitive description
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is still possible, as shown in the following diagram, but not in higher dimensions.
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\par
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As the input of simplices is sometimes cumbersome, we have defined a macro called {\tt code}
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to facilitate the input of the simplices.
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{\footnotesize\begin{verbatim}
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(defmacro code (arg) `(d (dlop-ext-int ,arg))) ==>
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CODE
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(? eml-mrp 3 (tnpr 2 (code '(0 1 2)) 1 (code '(3 4)) )) ==>
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----------------------------------------------------------------------{CMBN 3}
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<1 * <CrPr 0 0-1-2 2-1 3-4>>
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<-1 * <CrPr 1 0-1-2 2-0 3-4>>
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<1 * <CrPr 2 0-1-2 1-0 3-4>>
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------------------------------------------------------------------------------
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(? eml-mrp 6 (tnpr 2 (code '(0 3 5)) 4 (code '(0 1 2 4 5)) )) ==>
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----------------------------------------------------------------------{CMBN 6}
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<1 * <CrPr 3-2-1-0 0-3-5 5-4 0-1-2-4-5>>
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<-1 * <CrPr 4-2-1-0 0-3-5 5-3 0-1-2-4-5>>
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<1 * <CrPr 4-3-1-0 0-3-5 5-2 0-1-2-4-5>>
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<-1 * <CrPr 4-3-2-0 0-3-5 5-1 0-1-2-4-5>>
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<1 * <CrPr 4-3-2-1 0-3-5 5-0 0-1-2-4-5>>
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<1 * <CrPr 5-2-1-0 0-3-5 4-3 0-1-2-4-5>>
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<-1 * <CrPr 5-3-1-0 0-3-5 4-2 0-1-2-4-5>>
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<1 * <CrPr 5-3-2-0 0-3-5 4-1 0-1-2-4-5>>
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<-1 * <CrPr 5-3-2-1 0-3-5 4-0 0-1-2-4-5>>
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<1 * <CrPr 5-4-1-0 0-3-5 3-2 0-1-2-4-5>>
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<-1 * <CrPr 5-4-2-0 0-3-5 3-1 0-1-2-4-5>>
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<1 * <CrPr 5-4-2-1 0-3-5 3-0 0-1-2-4-5>>
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<1 * <CrPr 5-4-3-0 0-3-5 2-1 0-1-2-4-5>>
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<-1 * <CrPr 5-4-3-1 0-3-5 2-0 0-1-2-4-5>>
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<1 * <CrPr 5-4-3-2 0-3-5 1-0 0-1-2-4-5>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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\newpage
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%
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\vskip 0.40cm
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\centerline{\includegraphics{eml2.eps}}
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\vskip 0.40cm
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%
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The Alexander-Whitney homomorphism works in the reverse sense:
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{\footnotesize\begin{verbatim}
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(setf aw-mrp (aw (soft-delta-infinity)(soft-delta-infinity))) ==>
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[K12 Morphism (degree 0)]
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\end{verbatim}}
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The 2-simplices of the cartesian product $(0\, 1\, 2) \times (3\, 4)$ are (here, the degeneracy
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operators are in clear):
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{\footnotesize\begin{verbatim}
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<CrPr 0 (0 1) 1 (3 4)>
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<CrPr 1 (0 1) 0 (3 4)>
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<CrPr 0 (0 2) 1 (3 4)>
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<CrPr 1 (0 2) 0 (3 4)>
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<CrPr 0 (1 2) 1 (3 4)>
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<CrPr 1 (1 2) 0 (3 4)>
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<CrPr * (0 1 2) 1-0 (3)>
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<CrPr * (0 1 2) 1-0 (4)>
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--> <CrPr * (0 1 2) 0 (3 4)>
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<CrPr * (0 1 2) 1 (3 4)>
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\end{verbatim}}
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The last but one line of this list describes the triangle shown in dashed lines in the prism
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in the previous picture, joining the points $(0,3), (1,3), (2,4)$.
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Let us apply the homomorphism $\cal AW$:
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{\footnotesize\begin{verbatim}
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(? aw-mrp 2 (crpr 0 (code '(0 1 2)) 1 (code '(3 4)) )) ==>
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----------------------------------------------------------------------{CMBN 2}
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<1 * <TnPr 0-1 3-4>>
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<1 * <TnPr 0-1-2 4>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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We see that we obtain a kind of decomposition, up to a homotopy, of this triangle
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in terms of the two tensor products $(0\, 1\, 2) \otimes (4)$ (the ``upper'' triangle of
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the prism) and $(0\,1) \otimes (3\, 4)$ (the ``front'' rectangle of the prism).
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\par
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\medskip
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The composition ${\cal AW} \circ {\cal EML}$ is the identity in the tensor product of
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the two chain complexes.
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{\footnotesize\begin{verbatim}
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(setf g (tnpr 2 (code '(0 1 2)) 3 (code '(2 3 4 5)) )) ==>
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<TnPr 0-1-2 2-3-4-5>
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(? eml-mrp 5 g) ==>
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----------------------------------------------------------------------{CMBN 5}
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<1 * <CrPr 2-1-0 0-1-2 4-3 2-3-4-5>>
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<-1 * <CrPr 3-1-0 0-1-2 4-2 2-3-4-5>>
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<1 * <CrPr 3-2-0 0-1-2 4-1 2-3-4-5>>
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<-1 * <CrPr 3-2-1 0-1-2 4-0 2-3-4-5>>
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<1 * <CrPr 4-1-0 0-1-2 3-2 2-3-4-5>>
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<-1 * <CrPr 4-2-0 0-1-2 3-1 2-3-4-5>>
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<1 * <CrPr 4-2-1 0-1-2 3-0 2-3-4-5>>
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<1 * <CrPr 4-3-0 0-1-2 2-1 2-3-4-5>>
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<-1 * <CrPr 4-3-1 0-1-2 2-0 2-3-4-5>>
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<1 * <CrPr 4-3-2 0-1-2 1-0 2-3-4-5>>
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------------------------------------------------------------------------------
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(? aw-mrp *)
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----------------------------------------------------------------------{CMBN 5}
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<1 * <TnPr 0-1-2 2-3-4-5>>
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------------------------------------------------------------------------------
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(setf *tnpr-with-degrees* t) ==>
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T
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**
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----------------------------------------------------------------------{CMBN 5}
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<1 * <TnPr 2 0-1-2 3 2-3-4-5>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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But the composition ${\cal EML} \circ {\cal AW}$ is by no means the identity. This composition
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is related to the identity via the homotopy morphism $\Phi$. Let us consider the
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two following simple simplicial sets {\tt delta-0-1} and {\tt delta-2-3}, where the
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second is the simplicial set, model of the segment $[2,\,3]$. To build {\tt delta-2-3}, we
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have just changed the basis function, the base point and the comment in the system function {\tt soft-delta}.
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{\footnotesize\begin{verbatim}
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(setf delta-0-1 (soft-delta 1))
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[K13 Simplicial-Set]
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(setf delta-2-3 (build-smst
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:cmpr #'soft-delta-cmpr
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:basis #'(lambda(dmn)
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(case dmn
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(0 (list (d 4)(d 8)))
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(1 (list(d 12)))))
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:bspn (d 4)
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:face #'soft-delta-face
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:intr-dgnl #'soft-delta-dgnl :dgnl-strt :gnrt
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:intr-bndr #'soft-delta-bndr :bndr-strt :gnrt
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:orgn '(my-soft-delta-2-3)))
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[K18 Simplicial-Set]
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(show-structure delta-0-1 1) ==>
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Dimension = 0 :
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Vertices : (0 1)
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Dimension = 1 :
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Simplex : 0-1
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Faces : (<AbSm - 1> <AbSm - 0>)
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(show-structure delta-2-3 1) ==>
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Dimension = 0 :
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Vertices : (2 3)
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Dimension = 1 :
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Simplex : 2-3
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Faces : (<AbSm - 3> <AbSm - 2>)
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\end{verbatim}}
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Let us build the cartesian product of these two simplicial sets and let us list,
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in particular the
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0-simplices and the 1-simplices of this new simplicial set:
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{\footnotesize\begin{verbatim}
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(setf carre (crts-prdc delta-0-1 delta-2-3)) ==>
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[K23 Simplicial-Set]
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(show-structure carre 2) ==>
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Dimension = 0 :
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Vertices : (<CrPr - 0 - 2> <CrPr - 0 - 3> <CrPr - 1 - 2>
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<CrPr - 1 - 3>)
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Dimension = 1 :
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Simplex : <CrPr - 0-1 - 2-3>
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Faces : (<AbSm - <CrPr - 1 - 3>> <AbSm - <CrPr - 0 - 2>>)
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Simplex : <CrPr - 0-1 0 2>
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Faces : (<AbSm - <CrPr - 1 - 2>> <AbSm - <CrPr - 0 - 2>>)
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Simplex : <CrPr - 0-1 0 3>
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Faces : (<AbSm - <CrPr - 1 - 3>> <AbSm - <CrPr - 0 - 3>>)
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Simplex : <CrPr 0 0 - 2-3>
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Faces : (<AbSm - <CrPr - 0 - 3>> <AbSm - <CrPr - 0 - 2>>)
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Simplex : <CrPr 0 1 - 2-3>
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Faces : (<AbSm - <CrPr - 1 - 3>> <AbSm - <CrPr - 1 - 2>>)
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Dimension = 2 :
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Simplex : <CrPr 0 0-1 1 2-3>
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Faces : (<AbSm - <CrPr - 0-1 0 3>>
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<AbSm - <CrPr - 0-1 - 2-3>>
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<AbSm - <CrPr 0 0 - 2-3>>)
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Simplex : <CrPr 1 0-1 0 2-3>
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Faces : (<AbSm - <CrPr 0 1 - 2-3>>
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<AbSm - <CrPr - 0-1 - 2-3>>
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<AbSm - <CrPr - 0-1 0 2>>)
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\end{verbatim}}
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Let us take the first 1-simplex which corresponds to the diagonal of the square
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(realisation of the cartesian product) and
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let us build the three homomorphisms $\cal AW$, $\cal EML$ and $\Phi$:
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{\footnotesize\begin{verbatim}
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(setf diagonal (crpr 0 (code '(0 1)) 0 (code '(2 3)))) ==>
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<CrPr - 0-1 - 2-3>
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(setf aw-mrp (aw delta-0-1 delta-2-3)) ==>
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[K30 Morphism (degree 0)]
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(setf eml-mrp (eml delta-0-1 delta-2-3)) ==>
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[K31 Morphism (degree 0)]
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(setf phi-hmy (phi delta-0-1 delta-2-3)) ==>
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[K32 Morphism (degree 1)]
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\end{verbatim}}
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We see that
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$$ {\cal AW}((0\, 1), (2\, 3)) = (0\, 1)\otimes (3) + (0) \otimes (2\, 3),$$
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$$ {\cal EML}((0\, 1)\otimes (3) + (0) \otimes (2\, 3))= ((0\, 1), \eta_0 (3))
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+ (\eta_0 (0), (2\, 3))$$
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and that $\Phi$ applied to the diagonal returns the 2-simplex compatible
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with the homotopy.
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$$\Phi((0\, 1), (2\, 3))= (\eta_1(2\, 3),\eta_0(0\, 1)).$$
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This is illustrated by the following diagram (the 2-simplex is the shaded triangle).
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\vskip 0.50cm
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%
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\centerline{\includegraphics{eml3.eps}}
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%
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\vskip 0.50cm
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{\footnotesize\begin{verbatim}
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(? aw-mrp 1 diagonal) ==>
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----------------------------------------------------------------------{CMBN 1}
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<1 * <TnPr 0 0 1 2-3>>
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<1 * <TnPr 1 0-1 0 3>>
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------------------------------------------------------------------------------
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(? eml-mrp *) ==>
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----------------------------------------------------------------------{CMBN 1}
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<1 * <CrPr - 0-1 0 3>>
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<1 * <CrPr 0 0 - 2-3>>
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------------------------------------------------------------------------------
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(? phi-hmy 1 diagonal) ==>
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----------------------------------------------------------------------{CMBN 2}
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<-1 * <CrPr 0 0-1 1 2-3>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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Let us inspect now the reduction generated by the function {\tt ez} and
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verify on reasonably general combinations that the involved morphisms are coherent.
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{\footnotesize\begin{verbatim}
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(setf ez-rdc (ez (soft-delta-infinity)(soft-delta-infinity))) ==>
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[K34 Reduction]
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(inspect *) ==>
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REDUCTION @ #x36a55a = [K34 Reduction]
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0 Class --------> #<STANDARD-CLASS REDUCTION>
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1 ORGN ---------> <...>, a proper list with 3 elements
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2 IDNM ---------> fixnum 34 [#x00000088]
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3 H ------------> [K33 Morphism (degree 1)]
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4 G ------------> [K11 Morphism (degree 0)]
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5 F ------------> [K12 Morphism (degree 0)]
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6 BCC ----------> [K3 Chain-Complex]
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7 TCC ----------> [K6 Simplicial-Set]
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\end{verbatim}}
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With the comment slots we see that the morphism {\em f}, {\em g} and {\em h}
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are respectively {\em aw}, {\em eml} and {\em phi}:
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{\footnotesize\begin{verbatim}
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(orgn ez-rdc) ==>
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(EILENBERG-ZILBER [K1 Simplicial-Set][K1 Simplicial-Set])
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(orgn (f ez-rdc)) ==>
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(AW [K1 Simplicial-Set][K1 Simplicial-Set])
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(orgn (g ez-rdc)) ==>
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(EML [K1 Simplicial-Set][K1 Simplicial-Set])
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(orgn (h ez-rdc)) ==>
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(PHI [K1 Simplicial-Set][K1 Simplicial-Set])
|
|
|
|
(setf *bc* (cmbn 3 1 (tnpr 0 (code '(0)) 3 (code '(1 2 3 4)))
|
|
10 (tnpr 1 (code '(5 6)) 2 (code '(7 8 9)))
|
|
100 (tnpr 2 (code '(10 11 12)) 1 (code '(13 14)))
|
|
1000 (tnpr 3 (code '(15 16 17 18)) 0 (code '(19))))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
<1 * <TnPr 0 1-2-3-4>>
|
|
<10 * <TnPr 5-6 7-8-9>>
|
|
<100 * <TnPr 10-11-12 13-14>>
|
|
<1000 * <TnPr 15-16-17-18 19>>
|
|
------------------------------------------------------------------------------
|
|
\end{verbatim}}
|
|
\newpage
|
|
{\footnotesize\begin{verbatim}
|
|
(setf *tc*
|
|
(cmbn 3 1 (crpr 0 (code '(0 1 2 3)) 0 (code '(5 6 7 8)))
|
|
10 (crpr 0 (code '(0 1 2 3)) (dgop-ext-int '(2 0)) (code '(5 6)))
|
|
100 (crpr (dgop-ext-int '(2 1)) (code '(0 1)) 0 (code '(5 6 7 8)))
|
|
1000 (crpr (dgop-ext-int '(2 1)) (code '(0 1)) 1 (code '(5 6 7)))))
|
|
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
<1 * <CrPr - 0-1-2-3 - 5-6-7-8>>
|
|
<10 * <CrPr - 0-1-2-3 2-0 5-6>>
|
|
<100 * <CrPr 2-1 0-1 - 5-6-7-8>>
|
|
<1000 * <CrPr 2-1 0-1 0 5-6-7>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(pre-check-rdct ez-rdc) ==>
|
|
|
|
---done---
|
|
|
|
(check-rdct) ==>
|
|
|
|
*TC* =>
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
<1 * <CrPr - 0-1-2-3 - 5-6-7-8>>
|
|
<10 * <CrPr - 0-1-2-3 2-0 5-6>>
|
|
<100 * <CrPr 2-1 0-1 - 5-6-7-8>>
|
|
<1000 * <CrPr 2-1 0-1 0 5-6-7>>
|
|
------------------------------------------------------------------------------
|
|
|
|
*BC* =>
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
<1 * <TnPr 0 1-2-3-4>>
|
|
<10 * <TnPr 5-6 7-8-9>>
|
|
<100 * <TnPr 10-11-12 13-14>>
|
|
<1000 * <TnPr 15-16-17-18 19>>
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *TDD* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 1}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *BDD* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 1}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *DF-FD* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 2}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *DG-GD* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 2}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *ID-FG* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *ID-GF-DH-HD* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *HH* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 5}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *FH* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 4}
|
|
------------------------------------------------------------------------------
|
|
|
|
Checking *HG* = 0
|
|
Result:
|
|
----------------------------------------------------------------------{CMBN 4}
|
|
------------------------------------------------------------------------------
|
|
|
|
---done---
|
|
\end{verbatim}}
|
|
|
|
\section {Application to Homology}
|
|
|
|
Let us illustrate the Eilenberg-Zilber theorem\index{Eilenberg-Zilber theorem!application of} by the following example.
|
|
Let us consider the manifold $P^2{\R}\times S^3$. The theorem says that
|
|
$$H_n({\cal C}_*(P^2{\R}\times S^3)) \cong H_n({\cal C}_*(P^2{\R}) \otimes {\cal C}_*(S^3)).$$
|
|
{\footnotesize\begin{verbatim}
|
|
(setf p2 (moore 2 1)) ==> ;; Moore(2,1) generates the projectif plane.
|
|
|
|
[K1 Simplicial-Set]
|
|
|
|
(setf s3 (sphere 3)) ==>
|
|
|
|
[K6 Simplicial-Set]
|
|
\end{verbatim}}
|
|
The simplicial set {\tt p2-X-s3} is obtained from the cartesian product of
|
|
the two simplicial sets {\tt p2} and {\tt s3}.
|
|
{\footnotesize\begin{verbatim}
|
|
(setf p2-X-s3 (crts-prdc p2 s3)) ==>
|
|
|
|
[K11 Simplicial-Set]
|
|
\end{verbatim}}
|
|
Now, the chain complex {\tt p2-T-s3} is obtained from the tensor product of
|
|
the two chain complexes associated to both simplicial sets {\tt p2} and {\tt s3} (don't forget that the class
|
|
{\tt SIMPLICIAL SET} is a subclass of the class {\tt CHAIN COMPLEX}).
|
|
{\footnotesize\begin{verbatim}
|
|
(setf p2-T-s3 (tnsr-prdc p2 s3)) ==>
|
|
|
|
[K16 Chain-Complex]
|
|
\end{verbatim}}
|
|
Applying successively the function {\tt chcm-homology} (which computes directly the
|
|
homology groups without using a homotopy equivalence) on these chain
|
|
complexes, shows that the homology groups are effectively isomorphic,
|
|
but the method of the tensor product is much faster than the
|
|
method of the simple cartesian product, the number of generators being much more
|
|
smaller.
|
|
{\footnotesize\begin{verbatim}
|
|
(time(dotimes (i 6)(chcm-homology p2-X-s3 i))) ==>
|
|
|
|
Homology in dimension 0 :
|
|
|
|
Component Z
|
|
|
|
Homology in dimension 1 :
|
|
|
|
Component Z/2Z
|
|
|
|
Homology in dimension 2 :
|
|
|
|
Homology in dimension 3 :
|
|
|
|
Component Z
|
|
|
|
Homology in dimension 4 :
|
|
|
|
Component Z/2Z
|
|
|
|
Homology in dimension 5 :
|
|
|
|
---done---
|
|
|
|
; cpu time (non-gc) 880 msec user, 170 msec system
|
|
; cpu time (gc) 250 msec user, 0 msec system
|
|
; cpu time (total) 1,130 msec user, 170 msec system
|
|
; real time 2,563 msec
|
|
; space allocation:
|
|
; 43,812 cons cells, 64 symbols, 162,904 other bytes
|
|
|
|
(time(dotimes (i 6)(chcm-homology p2-T-s3 i))) ==>
|
|
|
|
Homology in dimension 0 :
|
|
|
|
Component Z
|
|
|
|
Homology in dimension 1 :
|
|
|
|
Component Z/2Z
|
|
|
|
Homology in dimension 2 :
|
|
|
|
Homology in dimension 3 :
|
|
|
|
Component Z
|
|
|
|
Homology in dimension 4 :
|
|
|
|
Component Z/2Z
|
|
|
|
Homology in dimension 5 :
|
|
|
|
---done---
|
|
|
|
; cpu time (non-gc) 130 msec user, 20 msec system
|
|
; cpu time (gc) 0 msec user, 0 msec system
|
|
; cpu time (total) 130 msec user, 20 msec system
|
|
; real time 326 msec
|
|
; space allocation:
|
|
; 1,897 cons cells, 0 symbols, 11,232 other bytes
|
|
\end{verbatim}}
|
|
The following example shows clearly the discrepancy between the length of
|
|
the basis of two objects built respectively by cartesian product and tensor product and
|
|
having the same homology groups.
|
|
{\footnotesize\begin{verbatim}
|
|
(setf s2 (sphere 2)) ==>
|
|
|
|
[K1 Simplicial-Set]
|
|
|
|
(setf s3 (sphere 3)) ==>
|
|
|
|
[K6 Simplicial-Set]
|
|
|
|
(setf s2Xs2Xs3 (crts-prdc (crts-prdc s2 s2) s3)) ==>
|
|
|
|
[K16 Simplicial-Set]
|
|
|
|
(orgn s2Xs2Xs3) ==>
|
|
|
|
(CRTS-PRDC [K23 Simplicial-Set] [K6 Simplicial-Set])
|
|
|
|
(setf s2Ts2Ts3 (tnsr-prdc (tnsr-prdc s2 s2) s3)) ==>
|
|
|
|
[K21 Chain-Complex]
|
|
|
|
(dotimes (i 7) (print(length(basis s2Xs2Xs3 i)))) ==>
|
|
|
|
1
|
|
0
|
|
3
|
|
22
|
|
138
|
|
390
|
|
480
|
|
|
|
(dotimes (i 7) (print(length(basis s2Ts2Ts3 i)))) ==>
|
|
|
|
1
|
|
0
|
|
2
|
|
1
|
|
1
|
|
2
|
|
0
|
|
\end{verbatim}}
|
|
|
|
\newpage
|
|
|
|
\subsection {Searching homology process for cartesian products}
|
|
|
|
When\index{searching homology!cartesian products} the {\tt search-efhm} method recognizes a cartesian product,
|
|
by means of the comment list (slot {\tt orgn}) of the object, it builds a homotopy equivalence where
|
|
the right bottom chain complex is created by the function {\tt ez}.
|
|
The process may be recursif as shown by the very definition of the method:
|
|
\vskip 0.35cm
|
|
{\footnotesize\begin{verbatim}
|
|
(defun LEFT-CRTS-PRDC-EFHM (smst1 smst2)
|
|
(declare (type simplicial-set smst1 smst2))
|
|
(the homotopy-equivalence
|
|
(build-hmeq
|
|
:lrdct (trivial-rdct (crts-prdc smst1 smst2))
|
|
:rrdct (ez smst1 smst2))))
|
|
|
|
(defmethod SEARCH-EFHM (smst (orgn (eql 'crts-prdc)))
|
|
(declare (type simplicial-set smst))
|
|
(the homotopy-equivalence
|
|
(cmps
|
|
(left-crts-prdc-efhm (second (orgn smst))
|
|
(third (orgn smst)))
|
|
(tnsr-prdc (efhm (second (orgn smst)))
|
|
(efhm (third (orgn smst)))))))
|
|
\end{verbatim}}
|
|
|
|
\subsection* {Lisp files concerned in this chapter}
|
|
|
|
{\tt eilenberg-zilber.lisp}, {\tt searching-homology.lisp}.
|
|
|