554 lines
18 KiB
TeX
554 lines
18 KiB
TeX
\chapter {Serre spectral sequence}
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\section {Introduction}
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This chapter is devoted to the {\tt Kenzo} implementation of the
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Serre spectral sequence\index{Serre spectral sequence}
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with the aim to compute the homology groups
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of the total space of a fibration. From the programming point of view,
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this method will be used according to our general scheme of searching homology
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(generic function {\tt search-efhm}) applied to an object of type {\tt fibration-total}.
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\section {The topological problem}
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Let ${\cal G}$ be a simplicial group (the fiber space), $B$, a $1$--reduced simplicial set (the base space) and
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$\tau$ (the fibration), a simplicial morphism of degree $-1$, $\tau: B \longrightarrow {\cal G}$.
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We know, from a previous chapter (fibrations) that the software {\tt Kenzo} knows how to build
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the total space of the fibration: $B \times_\tau {\cal G}$, which has the same simplices as
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$B \times {\cal G}$ and the same face operators except the last one, given by:
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$$\partial_n(b,g) = (\partial_n b, \tau(b).\partial_n g), \quad b\in B, g \in {\cal G},$$
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where the '{\tt .}' denotes the group operation in ${\cal G}$. This discrepancy between the respective face operators
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$\partial_n$ induces a discrepancy between the differential operators. We suppose now, that
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the base and fiber spaces are objects of effective homology type, i.e. there exists homotopy equivalences
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$$\diagram{
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& \widehat{B} & & & & \widehat{\cal G} & & \cr
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\rho_1 \swarrow\nearrow & & \searrow\nwarrow \rho_2 &\quad & \rho'_1\swarrow\nearrow & & \searrow\nwarrow\rho'_2 & \cr
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B & & EB & & {\cal G} & & EG \cr
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}$$
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where the chain complexes $EB$ and $EG$ are effective. The problem consists in finding a homotopy equivalence
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between $B \times_\tau {\cal G}$ and an effective chain complex noted $EB \widetilde{\otimes} EG$;
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the underlying graded module of $EB \widetilde{\otimes} EG$ is $EB \otimes EG$ but the differential
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is twisted in a rather complicated way. In fact, the chain complex $EB \widetilde{\otimes} EG$ is nothing
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but the Hirsh complex\footnote{{\bf Guy Hirsh.} {\em Sur les groupes d'homologie des espaces fibr\'es},
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Bulletin de la Soci\'et\'e Math\'ematique de Belgique, 1954, vol. 6, pp. 79-96.}.
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\subsection {The Ronald Brown reduction}
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We start from the Eilenberg-Zilber reduction (built in {\tt Kenzo} by the function {\tt ez}).
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$$
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\diagram{
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{{\cal C}_*(B \times {\cal G})} & \stackrel{h}{\longrightarrow} & {}^s{{\cal C}_*(B \times {\cal G})} \cr
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{\scriptstyle f} \downarrow \uparrow {\scriptstyle g} \cr
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{{\cal C}_*(B) \otimes {\cal C}_*({\cal G})} \cr
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}.
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$$
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But, in fact we are interested in the following one (the Ronald Brown reduction):
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$$
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\diagram{
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{{\cal C}_*(B \times_\tau {\cal G})} & \stackrel{h}{\longrightarrow} & {}^s{{\cal C}_*(B \times_\tau {\cal G})} \cr
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{\scriptstyle f} \downarrow \uparrow {\scriptstyle g} \cr
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{{\cal C}_*(B) \otimes_t {\cal C}_*({\cal G})} \cr
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}
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$$
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where the symbol $\otimes_t$ represents a {\rm twisted tensor product}, induced by the twisting
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operator $\tau$; this twisting operator is nothing but the Shih twisting cochain\footnote{{\bf Gugenheim.},
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{\em On the chain complex of a fibration}.
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Illinois Journal of Mathematics, 1972, vol. 16, pp. 398-414. } determined by Szczarba\footnote
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{{\bf R. H. Szczarba.}
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{\em The homology of twisted cartesian products}.
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Transactions of the American Mathematical Society,
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1961, vol. 100, pp. 197-216.}. Here, to obtain the Szczarba cochain, we apply the perturbation lemma (Ronnie Brown).
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In effect, writing
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$$ d_\tau = d + (d_\tau -d),$$
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one may consider that the differential $d_\tau$ of $B \times_\tau {\cal G}$
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is the differential $d$ of $B \times {\cal G}$ modified by the perturbation $d_\tau - d$.
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The perturbation morphism $d_\tau -d$ is deduced from the difference between both $\partial_n$ face operators.
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This being done, a perturbation may possibly be
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propagated along the Eilenberg-Zilber reduction (see the method {\tt add} applied to a reduction)
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and the perturbation lemma gives also, as a bonus, the perturbation to be applied to the dif\-fe\-ren\-ti\-al
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of $B \otimes {\cal G}$ to obtain the differential of $B \otimes_t {\cal G}$.
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\subsection {The Gugenheim algorithm}
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On the other hand, from the homotopy equivalences
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$$\diagram{
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& \widehat{B} & & & & \widehat{\cal G} & & \cr
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\rho_1 \swarrow\nearrow & & \searrow\nwarrow \rho_2 &\quad & \rho'_1\swarrow\nearrow & & \searrow\nwarrow\rho'_2 & \cr
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B & & EB & & {\cal G} & & EG \cr
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}$$
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it is possible to build their tensor product, i.e. the homotopy equivalence:
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$$\diagram{
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& \widehat{B}\otimes \widehat{\cal G} & \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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B\otimes {\cal G} & & EB \otimes EG \cr}
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$$
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This is done in {\tt Kenzo} by the method {\tt tnsr-prdc}.
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In fact, we are interested in the following one:
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$$\diagram{
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& \widehat{B}\widetilde{\otimes}\widehat{\cal G} & \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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B\otimes_t {\cal G} & & EB \widetilde{\otimes} EG \cr}
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$$
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where $\widehat{B}\widetilde{\otimes}\widehat{\cal G}$ and $EB \widetilde{\otimes} EG$ are appropriate
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strongly twisted tensor products determined by the perturbation lemma: this last homotopy equivalence
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is obtained by propagating over the previous one, the perturbation to be applied to the differential
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of $B \otimes {\cal G}$ to obtain the differential of $B \otimes_t {\cal G}$.
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The homotopy equivalence so obtained is called in {\tt Kenzo} the {\em right Serre homotopy equivalence}.
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\subsection {Assembling the puzzle}
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Let us build now the so-called {\em left Serre homotopy equivalence},
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$$\diagram{
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& B\times_\tau{\cal G} & \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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B\times_\tau {\cal G} & & B\otimes_t {\cal G} \cr}
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$$
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where the left reduction is the trivial one and the right reduction is
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the Brown reduction. We may then compose both Serre homotopy equivalences to obtain
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what we were looking for:
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$$\diagram{
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& B\times_\tau {\cal G} & & &\widehat{B}\otimes_t\widehat{\cal G} & & \cr
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\rho_1 \swarrow\nearrow & & \searrow\nwarrow \rho_2 & \rho'_1\swarrow\nearrow & & \searrow\nwarrow\rho'_2 & \cr
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B\times_\tau {\cal G} & &B\otimes_t {\cal G} &B\otimes_t {\cal G} & &EB \otimes_t EG \cr
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}$$
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\section {The functions for the Serre spectral sequence}
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The following functions are given for information purpose.
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\vskip 0.35cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt fibration-dtau-d-intr} {\em fibration} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the lisp function corresponding to the difference $d_\tau - d$, where $d$ is the differential
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in $B \times {\cal G}$ and $d_\tau$ the differential in $B \times_\tau {\cal G}$. This is an internal
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function and is used in the following one. \par}
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{\leftskip=5mm
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{\tt fibration-dtau-d} {\em fibration} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the morphism $d_\tau-d$, which is the perturbation morphism to be added to
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the differential $d$ of $B \times {\cal G}$ to obtain $d_\tau$,
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the differential in $B \times_\tau {\cal G}$. We recall that both simplicial sets have
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the same simplices. The source and the target of this morphism is the total space
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$B \times_\tau {\cal G}$ and the degree is $-1$. \par}
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{\leftskip=5mm
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{\tt Brown-reduction} {\em fibration} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return two values: a) the Brown reduction and b) the perturbation to be applied to the differential of the tensor
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product $B \otimes {\cal G}$ to obtain the differential of the twisted tensor
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product $B \otimes_t {\cal G}$. \par}
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{\leftskip=5mm
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{\tt left-Serre-efhm} {\em fibration} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the left Serre homotopy equivalence from the argument {\em fibration}. \par}
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{\leftskip=5mm
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{\tt right-Serre-efhm} {\em fibration} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the right Serre homotopy equivalence from the argument {\em fibration}. \par}
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}
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\section {Searching homology for fibration total spaces}
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The comment list of a {\tt Kenzo} object which is the total space of a fibration has the form
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\index{searching homology!fibration total space}
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{\tt (fibration-total {\em fibration})}. The {\tt search-efhm} method applied to a simplicial
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set having this kind of comment list, builds the composition of the left and right Serre homotopy
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equivalences. The right one needs the homotopy equivalences attached respectively to the
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base and fiber spaces. This may involve a recursive call of {\tt search-efhm}.
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If the base space or the fiber space
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are locally effective, the total space is locally effective and the method
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cannot compute the homology.
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{\footnotesize\begin{verbatim}
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(defmethod SEARCH-EFHM (smst (orgn (eql 'fibration-total)))
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(declare
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(type simplicial-set smst))
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(the homotopy-equivalence
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(fibration-total-efhm (second (orgn smst)))))
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(defun FIBRATION-TOTAL-EFHM (fibration)
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(declare (type fibration fibration))
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(the homotopy-equivalence
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(cmps (left-serre-efhm fibration)
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(right-serre-efhm fibration))))
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\end{verbatim}}
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\subsection* {Examples}
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We begin by some examples similar to examples that we have seen in the fibrations chapter.
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For the user, the delicate point is to write in Lisp a correct twisting operator. There is no check
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in this version of {\tt Kenzo} for the coherency of the twisting operator. In the first example
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we define a fibration $\tau: S^2 \rightarrow K({\Z},1)$ with $\tau(s2)= (2)$, then we build the
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total space of the fibration, namely $P^3{\R}$, and we compute some known homology groups.
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{\footnotesize\begin{verbatim}
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(setf sph2 (sphere 2)) ==>
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[K1 Simplicial-Set]
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(setf kz1 (k-z-1)) ==>
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[K6 Abelian-Simplicial-Group]
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(setf tw2 (build-smmr
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:sorc sph2
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:trgt kz1
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:degr -1
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:sintr #'(lambda (dmns gmsm)
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(unless (= dmns 2)
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(error "Dimension error for
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the twisting operator S2-->KZ1"))
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(absm 0 (list 2)))
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:orgn '(s2-tw-kz1))) ==>
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[K18 Fibration]
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(? tw2 2 's2) ==>
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<AbSm - (1)>
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(? tw2 0 '*) ==>
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Error: Dimension error for the twisting operator S2-->KZ1
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(setf p3r (fibration-total tw2)) ==>
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[K24 Simplicial-Set]
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(homology p3r 0 10) ==>
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Homology in dimension 0 :
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Component Z
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Homology in dimension 1 :
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Component Z/2Z
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Homology in dimension 2 :
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Homology in dimension 3 :
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Component Z
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Homology in dimension 4 :
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Homology in dimension 5 :
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Homology in dimension 5 :
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Homology in dimension 7 :
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Homology in dimension 8 :
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Homology in dimension 9 :
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--done--
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\end{verbatim}}
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Let us do the same kind of computations with $\tau(s2)= (5)$ and $\tau(s2)= (17)$.
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{\footnotesize\begin{verbatim}
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(setf tw5 (build-smmr
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:sorc sph2
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:trgt kz1
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:degr -1
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:sintr #'(lambda (dmns gmsm)
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(absm 0 (list 5)))
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:orgn '(s2-tw-kz1-5))) ==>
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[K110 Fibration]
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(setf total-5 (fibration-total tw5)) ==>
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[K111 Simplicial-Set]
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(homology total-5 0 5)
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Homology in dimension 0 :
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Component Z
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Homology in dimension 1 :
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Component Z/5Z
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Homology in dimension 2 :
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Homology in dimension 3 :
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Component Z
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Homology in dimension 4 :
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(setf tw17 (build-smmr
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:sorc sph2
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:trgt kz1
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:degr -1
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:sintr #'(lambda (dmns gmsm)
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(absm 0 (list 17)))
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:orgn '(s2-tw-kz1-17))) ==>
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[K165 Fibration]
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(setf total-17 (fibration-total tw17)) ==>
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[K166 Simplicial-Set]
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(homology total-17 0 4) ==>
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Homology in dimension 0 :
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Component Z
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Homology in dimension 1 :
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Component Z/17Z
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Homology in dimension 2 :
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Homology in dimension 3 :
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Component Z
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\end{verbatim}}
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\newpage
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Using the last total space ({\tt total-17}), it is instructive to enter in the details
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of the resulting homotopy equivalence that the system builds to compute the homology
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groups. First, the function {\tt echcm} selects the {\bf effective} chain complex of the homotopy
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equivalence, value of the slot {\tt efhm} of the simplicial set {\tt total-17}, and we may
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print some basis of this chain complex. In fact, all basis in dimension above $3$ are null.
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The presence of the symbol {\tt S1} is due to the fact that the effective chain complex
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used for the homology of $K({\Z},1)$ is the circle (see the classifying spaces chapter).
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{\footnotesize\begin{verbatim}
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(setf echcm (echcm total-17)) ==>
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[K205 Chain-Complex]
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(dotimes (i 6) (print (basis echcm i))) ==>
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(<TnPr * *>)
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(<TnPr * S1>)
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(<TnPr S2 *>)
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(<TnPr S2 S1>)
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NIL
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NIL
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\end{verbatim}}
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The resulting homotopy equivalence built by the system, may be summarized
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by the diagram:
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$$\diagram{
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&\widehat{S^2\times_\tau K({\Z},1)} & \cr
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f\swarrow\nearrow & & \searrow\nwarrow g \cr
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S^2\times_\tau K({\Z},1) & & [K205\, Chain\, Complex] \cr}
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$$
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where {\em f} and {\em g} are respectively the {\em descending} and {\em ascending}
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morphisms of the left and right reductions of the homotopy equivalence. We recall that
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these morphisms are obtained respectively by the functions {\tt lf} and {\tt rg}. We may
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compose these morphisms to create a morphism {\tt fg} from $[K205]$ to $S^2\times_\tau K({\Z},1)$ and
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see how it acts on the basis elements in dimension $1$ and $2$:
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{\footnotesize\begin{verbatim}
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(setf fg (cmps (lf (efhm total-17)) (rg (efhm total-17)))) ==>
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[K220 Morphism (degree 0)]
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(setf gen1 (first (basis echcm 1))) ==>
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<TnPr * S1>
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(? fg 1 gen1) ==>
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----------------------------------------------------------------------{CMBN 1}
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<-1 * <CrPr 0 * - (1)>>
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------------------------------------------------------------------------------
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(setf gen2 (first (basis echcm 2))) ==>
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<TnPr S2 *>
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(setf fg-gen2 (? fg 2 gen2)) ==>
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----------------------------------------------------------------------{CMBN 2}
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<-1 * <CrPr - S2 1-0 NIL>>
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<-1 * <CrPr 1-0 * - (1 1)>>
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<-1 * <CrPr 1-0 * - (1 2)>>
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<-1 * <CrPr 1-0 * - (1 3)>>
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<-1 * <CrPr 1-0 * - (1 4)>>
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<-1 * <CrPr 1-0 * - (1 5)>>
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<-1 * <CrPr 1-0 * - (1 6)>>
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<-1 * <CrPr 1-0 * - (1 7)>>
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<-1 * <CrPr 1-0 * - (1 8)>>
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<-1 * <CrPr 1-0 * - (1 9)>>
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<-1 * <CrPr 1-0 * - (1 10)>>
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<-1 * <CrPr 1-0 * - (1 11)>>
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<-1 * <CrPr 1-0 * - (1 12)>>
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<-1 * <CrPr 1-0 * - (1 13)>>
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<-1 * <CrPr 1-0 * - (1 14)>>
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<-1 * <CrPr 1-0 * - (1 15)>>
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<-1 * <CrPr 1-0 * - (1 16)>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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The boundary of this combination in $S^2\times_\tau K({\Z},1)$ is
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{\footnotesize\begin{verbatim}
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(? total-17 *)
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----------------------------------------------------------------------{CMBN 1}
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<-17 * <CrPr 0 * - (1)>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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to which corresponds in the effective chain complex, the boundary of the basis
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element in dimension $2$, $gen2=s2 \otimes *$:
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{\footnotesize\begin{verbatim}
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(? echcm 2 gen2) ==>
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----------------------------------------------------------------------{CMBN 1}
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<17 * <TnPr * S1>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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The printing of the $3$ faces of the $17$ simplices of the combination
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{\tt fg-gen2} shows the geometrical organisation of the twisted product:
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{\footnotesize\begin{verbatim}
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(dolist(iterm (cmbn-list fg-gen2))
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(dotimes (i 3)(print (face total-17 i 2 (gnrt iterm))))
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(terpri)) ==>
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<AbSm 0 <CrPr - * - NIL>>
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<AbSm 0 <CrPr - * - NIL>>
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<AbSm - <CrPr 0 * - (17)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (2)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (2)>>
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<AbSm - <CrPr 0 * - (3)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (3)>>
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<AbSm - <CrPr 0 * - (4)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (4)>>
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<AbSm - <CrPr 0 * - (5)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (5)>>
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<AbSm - <CrPr 0 * - (6)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (6)>>
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<AbSm - <CrPr 0 * - (7)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (7)>>
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<AbSm - <CrPr 0 * - (8)>>
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<AbSm - <CrPr 0 * - (1)>>
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<AbSm - <CrPr 0 * - (8)>>
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<AbSm - <CrPr 0 * - (9)>>
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<AbSm - <CrPr 0 * - (1)>>
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\end{verbatim}}
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\newpage
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{\footnotesize\begin{verbatim}
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<AbSm - <CrPr 0 * - (9)>>
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<AbSm - <CrPr 0 * - (10)>>
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|
<AbSm - <CrPr 0 * - (1)>>
|
|
|
|
<AbSm - <CrPr 0 * - (10)>>
|
|
<AbSm - <CrPr 0 * - (11)>>
|
|
<AbSm - <CrPr 0 * - (1)>>
|
|
|
|
<AbSm - <CrPr 0 * - (11)>>
|
|
<AbSm - <CrPr 0 * - (12)>>
|
|
<AbSm - <CrPr 0 * - (1)>>
|
|
|
|
<AbSm - <CrPr 0 * - (12)>>
|
|
<AbSm - <CrPr 0 * - (13)>>
|
|
<AbSm - <CrPr 0 * - (1)>>
|
|
|
|
<AbSm - <CrPr 0 * - (13)>>
|
|
<AbSm - <CrPr 0 * - (14)>>
|
|
<AbSm - <CrPr 0 * - (1)>>
|
|
|
|
<AbSm - <CrPr 0 * - (14)>>
|
|
<AbSm - <CrPr 0 * - (15)>>
|
|
<AbSm - <CrPr 0 * - (1)>>
|
|
|
|
<AbSm - <CrPr 0 * - (15)>>
|
|
<AbSm - <CrPr 0 * - (16)>>
|
|
<AbSm - <CrPr 0 * - (1)>>
|
|
|
|
<AbSm - <CrPr 0 * - (16)>>
|
|
<AbSm - <CrPr 0 * - (17)>>
|
|
<AbSm - <CrPr 0 * - (1)>>
|
|
\end{verbatim}}
|
|
\begin{center}
|
|
---------
|
|
\end{center}
|
|
We have seen that for the special case of loop spaces, there is a
|
|
canonical fibration and that the function {\tt twisted-crts-prdc}
|
|
builds the total space $X \times_\tau \Omega X$.
|
|
{\footnotesize\begin{verbatim}
|
|
(setf sph3 (sphere 3)) ==>
|
|
|
|
[K221 Simplicial-Set]
|
|
|
|
(setf tw3 (twisted-crts-prdc sph3)) ==>
|
|
|
|
[K243 Simplicial-Set]
|
|
|
|
(face tw3 3 3 (crpr 0 's3 7 +null-loop+)) ==>
|
|
|
|
<AbSm - <CrPr 1-0 * - <<Loop[S3]>>>>
|
|
\end{verbatim}}
|
|
It is of course possible to study, any other fibration, provided one
|
|
defines a correct fibration $\tau$. In the following example
|
|
the fibration $\tau: S^3 \longrightarrow \Omega S^3$, is defined
|
|
by
|
|
$\tau(s3)= s3^{-2},$ where $s3^{-2}$ is a word in the Kan simplicial version
|
|
of the first loop space of $S^3$. In Kenzo, this object is created by the
|
|
statement:
|
|
{\footnotesize\begin{verbatim}
|
|
(absm 0 (loop3 0 's3 -2)) ==>
|
|
|
|
<AbSm - <<Loop[S3\-2]>>>
|
|
|
|
(setf tws3 (build-smmr :sorc sph3
|
|
:trgt (loop-space sph3)
|
|
:degr -1
|
|
:sintr #'(lambda (dmns gmsm) (absm 0 (loop3 0 's3 -2)))
|
|
:orgn '(s3-tw-lps3))) ==>
|
|
|
|
[K248 Fibration]
|
|
|
|
(setf total (fibration-total tws3)) ==>
|
|
|
|
[K249 Simplicial-Set]
|
|
|
|
(homology total 0 7) ==>
|
|
|
|
Homology in dimension 0 :
|
|
|
|
Component Z
|
|
|
|
Homology in dimension 1 :
|
|
|
|
Homology in dimension 2 :
|
|
|
|
Component Z/2Z
|
|
|
|
Homology in dimension 3 :
|
|
|
|
Homology in dimension 4 :
|
|
|
|
Component Z/2Z
|
|
|
|
Homology in dimension 5 :
|
|
|
|
Homology in dimension 6 :
|
|
|
|
Component Z/2Z
|
|
\end{verbatim}}
|
|
\newpage
|
|
|
|
\subsection* {Lisp files concerned in this chapter}
|
|
|
|
{\tt serre.lisp}, {\tt searching-homology.lisp}.
|