532 lines
17 KiB
TeX
532 lines
17 KiB
TeX
\chapter {Computing homotopy groups}
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\section {Mathematical aspects}
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The\index{computing homotopy groups} method used here to compute the homotopy groups is known as the {\bf Whitehead tower}.
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We recall some mathematical points:
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\subsection* {1) The Hurewicz theorem.}
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{\bf Theorem.}
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{\em Let X be a $1$--connected space such that all its homology groups $H_r(X)$ are null
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for $0<r<n$, then $\pi_n(X)\simeq H_n(X)$.}
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\subsection* {2) The properties of the classifying spaces.}
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Let $X$ such that $\pi_r(X)=0$ for $0<r<n$. From now on, we shall denote $\pi_n(X)$ simply by $\pi$.
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Let us consider the universal coefficients exact sequence:
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$$ 0 \longrightarrow Ext(H_{n-1}(X), \pi) \longrightarrow H^n(X,\pi) \longrightarrow Hom(H_n(X),\pi) \longrightarrow 0.$$
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Here $H_{n-1}(X)=0$, so that a canonical isomorphism is given:
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$$H^n(X,\pi) \buildrel {\simeq} \over \longrightarrow Hom(H_n(X),\pi),$$
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but $H_n(X) \simeq \pi$ and this gives us a canonical element $h \in H^n(X, \pi)$
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which is the fundamental $n$--th cohomology class in this context. A cocycle $\chi$ representing $h$ is constructed as
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follows. In the chain complex
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$$ \cdots \rightarrow C_{n+1} \buildrel {d_{n+1}} \over \longrightarrow C_n
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\buildrel {d_n} \over \longrightarrow C_{n-1} \longrightarrow \cdots,$$
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the kernel $Z_n$ of $d_n$ is a summand of $C_n(X)$:
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$$ C_n = Z_n \oplus S$$
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and such a decomposition induces a projection $p:C_n(X) \longrightarrow Z_n$.
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Finally, we have also the canonical projection $p':Z_n \longrightarrow H_n=\pi$.
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Then, $\chi$ can be defined by $\chi = p' \circ p$ and $\chi$ {\em is a cocycle}. If another
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$p$ is chosen, another $\chi$ is obtained but the cohomology class
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is the same.
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\par
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Let us fix now a choice for the cocycle $\chi: C_n(X) \longrightarrow \pi$. This cocycle induces in turn
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a simplicial map:
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$$ \varphi_\chi: C_n(X) \longrightarrow K(\pi,n), $$
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where $K(\pi,n)$ is the classifying group $K(\pi_n(X),n)$ of the group $\pi_n(X)=\pi$.
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If $\chi'$ is another choice for the cocycle then $\varphi_{\chi'}$ is homotopic to $\varphi_\chi$.
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We recall that the space $K(\pi,n)$ has the following properties:
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\begin{enumerate}
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\item $H_r(K(\pi,n))=0$ for $0 < r < n$,
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\item $\pi_r(K(\pi,n))=0$ for $r \not= n$,
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\item $\pi_n(K(\pi,n))=H_n(K(\pi,n)= \pi$.
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\end{enumerate}
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On the other hand, a canonical fibration of base space $K(\pi, n)$ and fibre space
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$K(\pi, n-1)$ is deduced from a canonical twisting operator $$\tau: K(\pi, n) \longrightarrow K(\pi, n-1)$$
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$$\diagram{
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K(\pi, n-1) \cr
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\downarrow \cr
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K(\pi, n) \times_\tau K(\pi, n-1) \cr
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\downarrow \cr
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K(\pi,n) }
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$$
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So, if a simplicial map $\varphi_\chi: X \longrightarrow K(\pi,n)$ is given,
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it is then possible to construct a fibration of base space $X$ and fibre space $K(\pi, n-1)$,
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the twisting operator being then defined by: $\tau_\chi = \tau \circ \varphi_\chi $, according to the diagram:
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$$\diagram{
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K(\pi, n-1) & \buildrel {=} \over \longrightarrow & K(\pi, n-1) \cr
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\downarrow & & \downarrow \cr
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X'=X \times_{\tau_\chi} K(\pi, n-1) & \buildrel {\psi_\chi} \over \longrightarrow & K(\pi,n) \times_\tau K(\pi, n-1) \cr
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\downarrow & & \downarrow \cr
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X & \buildrel \varphi_\chi \over \longrightarrow & K(\pi,n) }
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$$
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\subsection* {3) The Serre exact sequence of the homotopy groups of fibrations.}
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Let us consider the fibration
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$$ F \quad \hookrightarrow\quad T=X \otimes_\tau F \hskip 1.2em \longrightarrow \hskip -1.2em \longrightarrow
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\quad X,$$
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The Serre exact sequence establishes a connection between the homotopy groups of the $3$ spaces,
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in any dimension:
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$$\cdots \longrightarrow \pi_{n+1}(F) \longrightarrow \pi_{n+1}(T) \longrightarrow \pi_{n+1}(X)
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\longrightarrow \pi_n(F) \longrightarrow$$
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$$\longrightarrow \pi_n(T) \longrightarrow \pi_n(X) \longrightarrow \pi_{n-1}(F) \longrightarrow \cdots $$
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In our special case, where the fibration is:
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$$ K(\pi, n-1) \quad \hookrightarrow\quad X' \hskip 1.2em \longrightarrow \hskip -1.2em \longrightarrow
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\quad X,$$
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the Serre sequence may be written:
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$$\cdots \longrightarrow \pi_{n+1}(K(\pi, n-1)) \longrightarrow \pi_{n+1}(X') \longrightarrow \pi_{n+1}(X)
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\longrightarrow $$
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$$ \longrightarrow \pi_n(K(\pi, n-1)) \longrightarrow \pi_n(X') \longrightarrow \pi_n(X)
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\longrightarrow \pi_{n-1}(K(\pi, n-1)) \longrightarrow 0. $$
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But we know that $\pi_i(K(\pi,n))=0$ for $i \not= n$ and that $\pi_n(K(\pi, n))=\pi,$
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so the exact sequence may be re-written:
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$$ 0 \rightarrow \pi_{n+1}(X') \rightarrow \pi_{n+1}(X) \rightarrow 0 \rightarrow \pi_n(X')
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\rightarrow \pi_n(X) \rightarrow \pi \rightarrow 0. $$
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The subsequence:
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$$ 0 \longrightarrow \pi_n(X') \longrightarrow \pi_n(X)
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\longrightarrow \pi \longrightarrow 0. $$
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is exact; furthermore, the cocycle $\chi$ is such that the connection morphism
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$\partial: \pi_n(X) \rightarrow \pi$ is the identity; so that
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we deduce that $ \pi_n (X')=0 $.
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\par
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On the other hand, from the exactness of the subsequence
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$$ 0 \longrightarrow \pi_{n+1}(X') \longrightarrow \pi_{n+1}(X)
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\longrightarrow 0 $$
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we deduce that $ \pi_{n+1}(X') \simeq \pi_{n+1}(X)$
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and more generally $ \pi_r(X') \simeq \pi_r(X)$ for $r \not= n.$
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In particular, $\pi_r(X')=0$ for $r \leq n+1$ and the Hurewicz theorem gives again:
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$$ H_{n+1}(X') \simeq \pi_{n+1}(X') \simeq \pi_{n+1}(X).$$
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So, if we know how to compute $H_{n+1}(X')$ then we have obtained $\pi_{n+1}(X).$
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\subsection* {4) The Whitehead tower}
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Let\index{Whitehead tower} us denote for a reason which will be clear in a moment the space $X'$ by
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$X^{(n+1)}$.
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Due to the properties of $X^{(n+1)}$, we may iterate the process,
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namely, build the following fibrations, where $\pi_{n+1}(X^{(n+1)})$ is denoted simply by
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$\pi'$:
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$$\diagram{
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K(\pi', n) &\buildrel {=} \over \longrightarrow & K(\pi', n) \cr
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\downarrow & & \downarrow \cr
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X^{(n+2)}=X^{(n+1)} \times_{\tau_\chi} K(\pi', n) & \buildrel {\psi_\chi} \over \longrightarrow &
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K(\pi',n+1) \times_\tau K(\pi', n) \cr
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\downarrow & & \downarrow \cr
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X^{(n+1)} & \buildrel \varphi_\chi \over \longrightarrow & K(\pi',n+1) }
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$$
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where $X^{(n+2)}= X^{(n+1)} \times_{\tau_\chi} K(\pi', n)$ has the property
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$$H_{n+2}(X^{(n+2)}) \simeq \pi_{n+2}(X^{(n+2)}) \simeq \pi_{n+2}(X^{(n+1)}) \simeq \pi_{n+2}(X). $$
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This construction is known as the Whitehead tower.
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\section {The functions for computing homotopy groups}
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\subsection* {An important remark}
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In this version of Kenzo, only the case where the first non--null
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homology group (in non--null dimension) is ${\Z}$ or ${\Z}/{2 {\Z}}$ can be processed; however if this
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homology group is a direct sum of several copies of ${\Z}$ or ${\Z}/{2 {\Z}}$, then the corresponding
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stage of the Whitehead tower may also be constructed step by step.
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\vskip 0.35cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt chml-clss} {\em chcm first} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return a ``fundamental'' cohomology class, more precisely a cocycle $\chi$ defining it.
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The first argument {\em chcm} must be a chain complex $C_*$ with effective homology; in particular,
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an effective chain complex $EC_*$ is a by--product of the machine object $C_*$. The second argument ``{\em first}''
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is an integer $n$, namely the first non--null dimension from which the chain complex
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{\em chcm} has a non--null homology group, which {\bf must be isomorphic to ${\Z}$} or ${\Z}/{2 {\Z}}$.
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The reader may be amazed that this argument ``{\em first}'' must be provided, since it is a consequence
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of the given $C_*$. But in fact, the same function {\tt chml-class} may also be used in different contexts, for example,
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the Postnikov tower; in such a case, the argument {\em first} is not redundant. The returned cocycle $\chi$
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is in any case a chain complex morphism $\chi: EC_* \longrightarrow {\Z}$, where ${\Z}$ is the unit chain complex
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created by the function {\tt z-chcm}. The degree of the morphism is $- n$.
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It is important to note that the chain complex involved in the source of the morphism
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is the {\bf effective} chain complex of the homotopy equivalence value of the slot {\tt efhm} of
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the object {\em chcm}. See the section {\em The general method for computing homology} in the Homology chapter.
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Finally, if $H_n(C_*)={\Z}/{2 {\Z}}$, the actual cohomology class hoped by the user is the composition
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$p \circ \chi$, where $p$ is the canonical projection ${\Z} \longrightarrow {\Z}/{2 {\Z}}$. But nevertheless,
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the {\tt chml-clss} lisp function returns $\chi$ and not $p \circ \chi$.
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\par}
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{\leftskip=5mm
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{\tt z-whitehead} {\em smst chml-clss} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return a fibration over the simplicial set {\em smst} (the first argument), canonically associated
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to the ``cohomology class'' {\em chml-clss} (the second argument). The simplicial set $X$, i.e. {\em smst}, is
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reduced; its homotopy groups $\pi_r(X)$ are null for $0 \leq r \leq n-1$. The first non null homotopy group
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$\pi_n(X)$ is assumed to be ${\Z}$, i.e. $\pi_n(X)= H_n(X)= {\Z}$. The previous function {\tt chml-clss}, in
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this situation, returns a cocycle $\chi$, which may be used as the second argument {\em chml-clss} of the function
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{\tt z-whitehead}. The integer $n$ is determined by the absolute value of the degree of the
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cohomology class {\em chcm-clss}. As explained in the previous section, a canonical fibration
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is induced by $\chi$ and it is this fibration which is returned by {\tt z-whitehead}. The slot {\tt sintr} of the
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fibration is set by the internal function {\tt z-whitehead-sintr} which builds
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in an efficient way the lisp function implementing the twisting operator $\tau \circ \varphi_\chi$.
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\par}
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{\leftskip=5mm
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{\tt z2-whitehead} {\em smst chcm-clss} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return a fibration over the simplicial set {\em smst} (the first argument), canonically associated
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to the ``cohomology class'' {\em chml-clss} (the second argument). The simplicial set $X$, i.e. {\em smst}, is
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reduced; its homotopy groups $\pi_r(X)$ are null for $0 \leq r \leq n-1$. The first non null homotopy group
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$\pi_n(X)$ is assumed to be ${\Z}/2{\Z}$, i.e. $\pi_n(X)= H_n(X)= {\Z}/2{\Z}$. The previous function {\tt chml-clss}, in
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this situation, returns a ``cocycle'' $\chi$, which may be used as the second argument {\em chml-clss} of the function
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{\tt z2-whitehead}. The integer $n$ is determined by the absolute value of the degree of the
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cohomology class {\em chcm-clss}. In this ${\Z}/2{\Z}$ case, $\chi$ is even not a cocycle. The actual cocycle is
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obtained by the composition $p \circ \chi$, $p$ being the canonical projection
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${\Z} \longrightarrow {\Z}/{2 {\Z}}$. But the user is not concerned by these technicalities, because the function
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{\tt z2-whitehead} makes itself the necessary conversion. The slot {\tt sintr} of the
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fibration is set by the internal function {\tt z2-whitehead-sintr} which builds
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in an efficient way the lisp function implementing the twisting operator $\tau \circ \varphi_\chi$.
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\par}
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}
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\subsection* {Examples}
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Let us retrieve some known facts about $S^3$, in particular $\pi_4(S^3) \simeq {\Z}/{2 {\Z}}$.
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We follow the theoritical method exposed above, namely build the fibration
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$$ K({\Z},2) \quad \hookrightarrow\quad S^3 \times_{\tau_\chi} K({\Z},2) \hskip 1.2em
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\longrightarrow \hskip -1.2em \longrightarrow \quad S^3,$$
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and get the homology group in dimension $4$ of the total space.
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{\footnotesize\begin{verbatim}
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(setf s3 (sphere 3)) ==>
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[K1 Simplicial-Set]
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(homology s3 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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Homology in dimension 2 :
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Homology in dimension 3 :
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Component Z
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---done---
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(setf s3-chml-clss (chml-clss s3 3)) ==>
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[K12 Cohomology-Class (degree 3)]
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(setf s3-fibr (z-whitehead s3 s3-chml-clss)) ==>
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[K37 Fibration]
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(setf s3-total (fibration-total s3-fibr)) ==>
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[K43 Simplicial-Set]
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(homology s3-total 0 6)
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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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Homology in dimension 2 :
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Homology in dimension 3 :
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Homology in dimension 4 :
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Component Z/2Z
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Homology in dimension 5 :
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---done---
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\end{verbatim}}
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Let us show a similar example with the space $Moore(2,4)$. As $H_4 \simeq {\Z}/{2 {\Z}}$, the function
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for building the fibration is {\tt z2-whitehead}.
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We verify that $\pi_5(Moore(2,4))\simeq {\Z}/{2 {\Z}}$
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{\footnotesize\begin{verbatim}
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(setf m24 (moore 2 4)) ==>
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[K1 Simplicial-Set]
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(show-structure m24 6) ==>
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Dimension = 0 :
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Vertices : (*)
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Dimension = 1 :
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Dimension = 2 :
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Dimension = 3 :
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Dimension = 4 :
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Simplex : M4
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Faces : (<AbSm 2-1-0 *> <AbSm 2-1-0 *> <AbSm 2-1-0 *>
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<AbSm 2-1-0 *> <AbSm 2-1-0 *>)
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Dimension = 5 :
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Simplex : N5
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Faces : (<AbSm - M4> <AbSm 3-2-1-0 *> <AbSm - M4> <AbSm 3-2-1-0 *>
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<AbSm 3-2-1-0 *> <AbSm 3-2-1-0 *>)
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Dimension = 6 :
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NIL
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(homology m24 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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Homology in dimension 2 :
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Homology in dimension 3 :
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Homology in dimension 4 :
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Component Z/2Z
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---done---
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(setf m24-chml-clss (chml-clss m24 4)) ==>
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[K12 Cohomology-Class (degree 4)]
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(setf m24-fibr (z2-whitehead m24 m24-chml-clss)) ==>
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[K49 Fibration]
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(setf m24-total (fibration-total m24-fibr)) ==>
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[K55 Simplicial-Set]
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(homology m24-total 0 6) ==>
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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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Homology in dimension 2 :
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Homology in dimension 3 :
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Homology in dimension 4 :
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Homology in dimension 5 :
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Component Z/2Z
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---done---
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\end{verbatim}}
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We may even verify that, up to $9$, the homotopy groups of the classifying group $K({\Z},5)$, for instance,
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are null except $\pi_5$. This is a far-fetched method to verify this well known result, but
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it proves that the software is coherent.
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{\footnotesize\begin{verbatim}
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(setf k5 (k-z 5)) ==>
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[K49 Abelian-Simplicial-Group]
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(setf k5-chml-clss (chml-clss k5 5)) ==>
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[K576 Cohomology-Class (degree 5)]
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(setf k5-fibr (z-whitehead k5 k5-chml-clss)) ==>
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[K579 Fibration]
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(setf k5-total (fibration-total k5-fibr)) ==>
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[K580 Kan-Simplicial-Set]
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(homology k5-total 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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Homology in dimension 2 :
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Homology in dimension 3 :
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Homology in dimension 4 :
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Homology in dimension 5 :
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Homology in dimension 6 :
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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 show now the iteration of the process, to get for instance
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$\pi_5(S^3)$ and $\pi_6(S^3)$:
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{\footnotesize\begin{verbatim}
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(setf s3 (sphere 3)) ==>
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[K1 Simplicial-Set]
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(setf s3-chml-clss (chml-clss s3 3)) ==>
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[K12 Cohomology-Class (degree 3)]
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(setf s3-fibration (z-whitehead s3 s3-chml-clss)) ==>
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[K37 Fibration]
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(setf s3-4 (fibration-total s3-fibration)) ==>
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[K43 Simplicial-Set]
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(homology s3-4 4) ==>
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Homology in dimension 4 :
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Component Z/2Z
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---done---
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(setf s3-4-chml-clss (chml-clss s3-4 4)) ==>
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[K253 Cohomology-Class (degree 4)]
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(setf s3-4-fibration (z2-whitehead s3-4 s3-4-chml-clss)) ==>
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[K292 Fibration]
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(setf s3-5 (fibration-total s3-4-fibration)) ==>
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[K298 Simplicial-Set]
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(homology s3-5 5) ==>
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Homology in dimension 5 :
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Component Z/2Z
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---done---
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(setf s3-5-chml-clss (chml-clss s3-5 5)) ==>
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[K609 Cohomology-Class (degree 5)]
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(setf s3-5-fibration (z2-whitehead s3-5 s3-5-chml-clss)) ==>
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[K624 Fibration]
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(setf s3-6 (fibration-total s3-5-fibration)) ==>
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[K630 Simplicial-Set]
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(homology s3-6 6) ==>
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Component Z/12Z
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\end{verbatim}}
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An interesting example is given by the real projective space $P^\infty{\R}/P^1{\R}$ which may be built in
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{\tt Kenzo} by the function {\tt r-proj-space}. We list its first homotopy groups.
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{\footnotesize\begin{verbatim}
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(setf x (r-proj-space 2)) ==>
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[K1 Simplicial-Set]
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(homology x 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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Homology in dimension 2 :
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Component Z
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Homology in dimension 3 :
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Component Z/2Z
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Homology in dimension 4 :
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Homology in dimension 5 :
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Component Z/2Z
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Homology in dimension 6 :
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Homology in dimension 7 :
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Component Z/2Z
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Homology in dimension 8 :
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Homology in dimension 9 :
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Component Z/2Z
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---done---
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\end{verbatim}}
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Using the machinery described in this chapter, we find that the total space of the fibration
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has the same homology groups as $S^3$. It has been effectively proved that this space is homotopic
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to $S^3$.
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{\footnotesize\begin{verbatim}
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(setf ch (chml-clss x 2)) ==>
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[K12 Cohomology-Class (degree 2)]
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(setf f2 (z-whitehead x ch)) ==>
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[K25 Fibration]
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(setf x3 (fibration-total f2)) ==>
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[K31 Simplicial-Set]
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(homology x3 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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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 6 :
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Homology in dimension 7 :
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---done---
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\end{verbatim}}
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\subsection* {Lisp files concerned in this chapter}
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{\tt whitehead.lisp}, {\tt smith.lisp}.
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