419 lines
12 KiB
TeX
419 lines
12 KiB
TeX
\chapter {Fibrations}
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In {tt Kenzo}, the fibration\index{fibrations} theory is applied in the simplicial frame
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and limited to the {\em simplicial principal fiber spaces}.
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\section {Notion of fibration}
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Let $B$ a simplicial set and $\cal G$ a simplicial group. A {\em (right--hand) simplicial fibration} is
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defined by a {\bf twisting operator}\index{fibration!twisting operator}, $\tau : B \longrightarrow {\cal G}$,
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of degree $-1$.
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For any dimension $n$ and for any element $b \in B_n$, the face and degeneracy operators must
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satisfy the following relations:
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\begin{eqnarray*}
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\partial_i(\tau b) & = & \tau(\partial_i b), \qquad i < n-1, \\
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\partial_{n-1}(\tau b) & = & [\tau(\partial_n b)]^{-1}.\tau(\partial_{n-1} b), \\
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\eta_i(\tau b) & = & \tau (\eta_i b), \qquad i \leq n-1, \\
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e_n & = & \tau(\eta_n b).
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\end{eqnarray*}
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where $e_n$ is the neutral element of ${\cal G}_n$.
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\vskip 0.30cm
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Now, the twisting operator $\tau$ defines a fibration of base space $B$, of fiber space $\cal G$
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and of total space $B \times_{\tau} {\cal G}$, whose simplicial components are:
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$$ (B \times_{\tau} {\cal G})_n = (B \times {\cal G})_n = (B_n \times {\cal G}_n), $$
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and whose face operators are defined,
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in any dimension $n$ and for any pair $(b,g),\, b \in B_n,\, g \in {\cal G}_n$ by:
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\begin{eqnarray*}
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\partial_i(b,g) & = & (\partial_i b, \partial_i g), \quad i < n, \\
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\partial_n(b,g) & = & (\partial_n b, \tau(b).\partial_n g).
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\end{eqnarray*}
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As it is well known, there are $4$ natural choices for such a definition. The action of the
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group can be from the right or from the left and the critical index can be the last one ($n$) or
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the first one ($0$). In {\tt Kenzo}, the implementor has chosen the non--usual one (i.e. action
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on the right hand and critical index, the last one), leading to the
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Szczarba formulas, the best choice in the case of the loop spaces.
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\section {Building a twisting operator}
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To build a twisting operator, one uses the function {\tt build-smmr} designed as a convenient tool
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to build a simplicial morphism. The source (keyword {\tt :sorc}) is the base space, a {\bf reduced} simplicial
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set, the target (keyword {\tt :trgt}) is the fiber space, a simplicial group. The degree is $-1$ and
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the morphism (keyword {\tt :sintr}) is the internal lisp function implementing $\tau$. See the example
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below, where $\tau$ is a simple twisting operator, applying in particular the only (geometrical) simplex
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of degree $2$ of $S^2$ upon the bar object $[1] \in (K({\Z}_2,1))_1$. \par
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In {\tt Kenzo}, the type {\tt FIBRATION} is defined. An object of this type must be of type
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{\tt SIMPLICIAL-MRPH}, the degree of the corresponding morphism being $-1$. To apply the morphism,
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one uses the well known macro {\tt ?} or the special macro {\tt tw-a-sintr3}.
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\vskip 0.30cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt ?} {\em twop dmns gmsm} \hfill {\em [Macro]} \par}
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{\leftskip=15mm
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Apply the twisting operator {\em twop} to the geometrical simplex {\em gmsm} in dimension
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{\em dmns}. Of course, {\tt ?} works as well with a combination of geometric simplices
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as in {\tt (? {\em twop cmbn})}. \par}
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{\leftskip=5mm
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{\tt tw-a-sintr3} {\em sintr dmns absm bspn} \hfill {\em [Macro]} \par}
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{\leftskip=15mm
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Apply the internal lisp function {\em sintr} implementing a twisting o\-pe\-ra\-tor to the
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abstract simplex {\em absm} in dimension {\em dmns}. The base point {\em bspn} of the
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target simplicial set (a simplicial group) must be given, since the function may return the neutral
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element in dimension $dmns - 1$. \par}
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}
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\newpage
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\section {Constructing the total space}
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The total space\index{fibration!total space} associated to a twisting operator is constructed
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by the function {\tt fibration-total}.
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\vskip 0.30cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt fibration-total} {\em fibration} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the simplicial set, total space of the fibration. This function
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extracts from the simplicial morphism {\em fibration} (a previously defined twisting operator),
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the base space (a simplicial set)
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and the fiber space (a simplicial group) and constructs
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the twisted cartesian product of these simplicial sets.
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The face function is built using the internal function {\tt fibration-total-face}.
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The non--degenerate simplices of the total space are coded exactly as those of
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the non--twisted product ($ B \times {\cal G}$) obtained by the lisp call
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{\tt (crts-prdc {\em base} {\em fiber})}. This non-twisted cartesian product is saved
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into the slot {\tt grmd} (graded module) of the instance object.
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If the base space and the fiber space are both of type {\tt KAN}, then the filling function is computed
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by the internal function {\tt fibration-kfll} and the function {\tt fibration-total}
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returns an object of type {\tt KAN}. \par}
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}
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\subsection* {Examples}
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{\footnotesize\begin{verbatim}
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(setf s2 (sphere 2)) ==>
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[K1 Simplicial-Set]
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(setf kz2 (k-z2-1)) ==>
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[K6 Abelian-Simplicial-Group]
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(setf tw2 (build-smmr
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:sorc s2
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:trgt kz2
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:degr -1
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:sintr #'(lambda (dmns gmsm) (absm 0 1))
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:orgn '(s2-tw-kz2))) ==>
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[K18 Fibration]
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\end{verbatim}}
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\newpage
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{\footnotesize\begin{verbatim}
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(inspect tw2) ==>
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SIMPLICIAL-MRPH @ #x4a54c2 = [K18 Fibration]
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0 Class --------> #<STANDARD-CLASS SIMPLICIAL-MRPH>
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1 ORGN ---------> (S2-TW-KZ2), a proper list with 1 element
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2 IDNM ---------> fixnum 18 [#x00000048]
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3 RSLTS --------> simple T vector (15) = #(#() #() #() ...)
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4 ?-CLNM -------> fixnum 0 [#x00000000]
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5 ???-CLNM -----> fixnum 0 [#x00000000]
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6 STRT ---------> The symbol :GNRT
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7 INTR ---------> The symbol NIL
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8 DEGR ---------> fixnum -1 [#xfffffffc]
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9 TRGT ---------> [K6 Abelian-Simplicial-Group]
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10 SORC ---------> [K1 Simplicial-Set]
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11 SINTR --------> #<Interpreted Function (unnamed) @ #x4a545a>
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\end{verbatim}}
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The following statement show how to apply the twisting operator to a
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{\em geometrical} simplex. But if we have an abstract simplex, one must
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use the special function {\tt tw-a-sintr3}.
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{\footnotesize\begin{verbatim}
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(? tw2 2 's2) ==>
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<AbSm - 1>
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(tw-a-sintr3 (sintr tw2) 3 (absm 1 's2) nil) ==>
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<AbSm 0 1>
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(tw-a-sintr3 (sintr tw2) 3 (absm 7 '*) nil) ==>
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<AbSm 1-0 NIL>
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(dotimes (i 3) (print (tw-a-sintr3 (sintr tw2) 3
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(absm i 's2) nil))) ==>
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<AbSm - 1>
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<AbSm 0 1>
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<AbSm 1 1>
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\end{verbatim}}
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Let us build the total space of the fibration (a simplicial set). We show some basis and compute
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some homology groups. The method to compute the homology groups will be explained in a following
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chapter.
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{\footnotesize\begin{verbatim}
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(setf tot-spc2 (fibration-total tw2)) ==>
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[K24 Simplicial-Set]
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(inspect tot-spc2) ==>
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SIMPLICIAL-SET @ #x4a87f2 = [K24 Simplicial-Set]
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0 Class --------> #<STANDARD-CLASS SIMPLICIAL-SET>
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1 ORGN ---------> <...>, a proper list with 2 elements
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2 IDNM ---------> fixnum 24 [#x00000060]
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3 EFHM ---------> The symbol :--UNBOUND--
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4 GRMD ---------> [K19 Simplicial-Set]
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5 DFFR ---------> [K25 Morphism (degree -1)]
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6 BSGN ---------> <CrPr - * - 0>, a dotted list with 3 elements
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7 BASIS --------> #<Closure (FLET CRTS-PRDC-BASIS RSLT) @ #x4a735a>
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8 CMPR ---------> #<Closure (FLET CRTS-PRDC-CMPR RSLT) [#'SPHERE-CMPR] @ #x4a732a>
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9 CPRD ---------> [K28 Morphism (degree 0)]
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10 FACE ---------> #<Closure (FLET FIBRATION-TOTAL-FACE RSLT) @ #x4a8752>
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(basis tot-spc2 0) ==>
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(<CrPr - * - 0>)
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(basis tot-spc2 1) ==>
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(<CrPr 0 * - 1>)
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(basis tot-spc2 2) ==>
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(<CrPr - S2 - 2> <CrPr - S2 0 1> <CrPr - S2 1 1> <CrPr - S2 1-0 0> <CrPr 1-0 * - 2>)
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(basis tot-spc2 3) ==>
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(<CrPr 0 S2 - 3> <CrPr 0 S2 1 2> <CrPr 0 S2 2 2> <CrPr 0 S2 2-1 1> <CrPr 1 S2 - 3>
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<CrPr 1 S2 0 2> <CrPr 1 S2 2 2> <CrPr 1 S2 2-0 1> <CrPr 2 S2 - 3> <CrPr 2 S2 0 2>
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<CrPr 2 S2 1 2> <CrPr 2 S2 1-0 1> <CrPr 2-1-0 * - 3>)
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(homology tot-spc2 0 8) ==>
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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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---done---
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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/4Z
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Homology in dimension 4 :
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---done---
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Homology in dimension 5 :
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Component Z/4Z
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Homology in dimension 6 :
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---done---
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Homology in dimension 7 :
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Component Z/4Z
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\end{verbatim}}
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In the following example, we take $K({\Z},1)$ and build a similar twisting o\-pe\-ra\-tor. Note
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that in $(K({\Z},1))_1$, the bar object $[1]$ is represented as {\tt (1)} whereas in $K({\Z}_2,1)$ it
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is {\tt 1}. In this case, it is known that the total space is a model of the sphere $S^3$
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(Hopf fibration - see below).
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{\footnotesize\begin{verbatim}
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(setf s2 (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 tw1 (build-smmr
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:sorc s2
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:trgt kz1
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:degr -1
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:sintr #'(lambda (dmns gmsm) (absm 0 (list 1)))
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:orgn '(s2-tw-kz1))) ==>
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[K18 Fibration]
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(setf tot-spc1 (fibration-total tw1)) ==>
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[K24 Simplicial-Set]
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(homology tot-spc1 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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---done---
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Homology in dimension 2 :
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---done---
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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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---done---
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Homology in dimension 5 :
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---done---
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\end{verbatim}}
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We may also obtain the boundary and faces of elements of the total space:
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{\footnotesize\begin{verbatim}
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(setf elem (crpr 0 's2 0 '(1 2))) ==>
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<CrPr - S2 - (1 2)>
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(? tot-spc1 2 elem) ==>
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----------------------------------------------------------------------{CMBN 1}
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<2 * <CrPr 0 * - (2)>>
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<-1 * <CrPr 0 * - (3)>>
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------------------------------------------------------------------------------
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(? tot-spc1 *) ==>
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----------------------------------------------------------------------{CMBN 0}
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------------------------------------------------------------------------------
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(dotimes (i 3) (print (face tot-spc1 i 2 elem))) ==>
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<AbSm - <CrPr 0 * - (2)>>
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<AbSm - <CrPr 0 * - (3)>>
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<AbSm - <CrPr 0 * - (2)>>
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\end{verbatim}}
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\newpage
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The system {\tt Kenzo} provides the function {\tt hopf} to build
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Hopf fibrations\index{fibrations! of Hopf}. The lisp definition is the following:
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{\footnotesize\begin{verbatim}
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(defun HOPF (n)
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(declare (fixnum n))
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(the simplicial-mrph
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(build-smmr
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:sorc (sphere 2)
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:trgt (k-z-1)
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:degr -1
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:sintr (hopf-sintr n)
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:orgn `(hopf ,n))))
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(defun HOPF-SINTR (n)
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(declare (fixnum n))
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(flet ((rslt (dmns gmsm)
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(declare (ignore dmns gmsm))
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(if (zerop n)
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(absm 1 +empty-list+)
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(absm 0 (list n)))))
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(the sintr #'rslt)))
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\end{verbatim}}
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In the definition of the lisp function for the slot {\tt sintr}, we see that the simplicial morphism
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applies in particular the only geometric simplex in dimension $2$ of $S^2$ onto the bar object
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$[n] \in (K({\Z},1))_1$. If $n=0$, this is $\eta_0 [\ ]$, i.e. the $0$--degeneracy of the base point
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{\tt NIL}. We give some examples of some fibrations and of the corresponding total spaces in function of the
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parameter {\tt n}. First we generate $4$ fibrations and apply the twisting operator
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on the geometrical simplex {\tt s2}.
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{\footnotesize\begin{verbatim}
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(setf h1 (hopf 1) h2 (hopf 2) h3 (hopf 3) h10 (hopf 10))
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(mapcar #'(lambda (h) (funcall (sintr (eval h)) 2 's2)) '(h1 h2 h3 h10)) ==>
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(<AbSm - (1)> <AbSm - (2)> <AbSm - (3)> <AbSm - (10)>)
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\end{verbatim}}
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Then we generate the corresponding total space and get some homology groups.
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{\footnotesize\begin{verbatim}
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(setf t1 (fibration-total h1) t2 (fibration-total h2)
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t3 (fibration-total h3) t10 (fibration-total h10))
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(homology t1 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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---done---
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Homology in dimension 2 :
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---done---
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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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---done---
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......... all next groups null .....
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(homology t2 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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---done---
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Homology in dimension 3 :
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Component Z
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---done---
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......... all next groups null .....
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(homology t3 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/3Z
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Homology in dimension 2 :
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---done---
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Homology in dimension 3 :
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Component Z
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(homology t10 1) ==>
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Homology in dimension 1 :
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Component Z/10Z
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\end{verbatim}}
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\subsection* {Lisp file concerned in this chapter}
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{\tt fibrations.lisp}.
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