Table Of ContentSimplicial and semisimplicial complexes
M. G. Barratt , 20 xii 1956.
1. Introduction.
The realiza,tion IBI of a CSS (= complete semisimplicial)
co!ILPlex B ([l] ; notation as in [7])has been defined in (3,4,6].
The definition in (6], where degeneracies are taken into account,
is followed here.
..
Theorem 1. ~ f~j has a simplicial subdivision.
If' di!lgeneracies are not taken into account, . IS d2 BI is,
,obviously simplicial (Sd is defined in (2-3}). Such rea.l..izationS!
•I . . ,··
};\ave disadv~tages~ ·e.g. IAI )c' IBI =J IA i.: Bl.
f,
Let R: (RB= IBI, Rf= Ir!) as in [6] be
~.
the realization functor.
Definition. A division functor on a (sub}category of CSS complexes
and maps to the category of CSS complexes and maps is a functor D
1
admitting natural transformations
~: D -+ 1, o;, RD -+ R,
1
Tb.at is, if fz B -+C, then there are commutative diagrams
°B
~
DB > B IDBI :> IBI
J
tDf ~f ; ·,·,ijDrl o:c lfl
~c
DC >C IDCI :> lcl
- - - - - - - - - - - - - - - .,. ------
1The author was ,partially SUJJPu.rtecJby Al.r Force Contract AF 18(600~•1494
during the period when the work on this paper was being done.
2
such that Ct_s: jDBj -+ IBI is a homeomorphism, and n_a ~ l~I,
I a I XI ) Fr I
there being a homotopy A such that · A ( c= if
Ct_s( I a I) c !'fl , ( 0'€ DB, T € B; II I ::: Clc:lsure of the realization
of~). Such functors·include the barycentric subdivision functor
f.1
Sd ( ( 2. 3) below) and the reverse nerve functor N · (§2) restricted
to the image of Sd. Therefore 1 is contained in the more
Tb.eo~em
explicit
6 .
Theorem 2 N Sd is a division functor from the category of css
complexes to the subcategory of simplicial complexes.
The proof that Sd is a division f'unctor is (fairly)straightforward.
The key to Theorem 1 lies in showing that the closed cells of ISdBI
6
are eiements. N SdB is isomorphic to the complex o?ta!ned by
starring the cells of !SdBI from internal points, in order of in-
creasing dimension.
The author is indebted to helpf'ul discussions with John Milnor,
who discovered Theorem 1 independently.
2. Nerve and Star Functors.
.
The nerve functors apply the formal process of subdivision of
simplicial complexes (see, for example,[5] ) to CSS complexes; the
star functors originate in Alexander's starring operation.
Definition. Tb.e nerve functor N (reverse nerve functor N6 )
assigns a simplicial complex NB (N6 B) to a CSS complex B· the
'
vertices are the nondegenerate sirnplices of B, and the p-simplices
3
=
(p 0,1,2, ••• ). are sequences (a(o)'"""' o(p)) such that
CJ( ) < • . . < CJ( ) (o(o) > ..• :::, o(p)~
0 - - p,
where a< T means that a is a fac·e of T • A map fi B -+ C
f
induces a transformation of the nondegenerate simplices of B
=
to the nondegenerate simplices of C such that fa fa or a
/::,.
degeneracy of fa • 'Ihen Nf (N f) is the map such that
Clearly, if B is a simplicial complex, NB is isomorphic to the
usual barycentric subdivision of B (c.f. (5] ).
Remark. '!here are (in general, not very useful) maps
(2.1) ')l.i NB -+ B,
defined on the vertices (a) by ')I.( a) = last vertex of a; A. t:,. ( o) = first
/::,. .
vertex of a, and such that ')l.~cr(o}'···iO'(p))' ')I. (a(o)'"""'o(p))
are in the least subcomplexes of B containing
respectively.
'Ihe face and degeneracy operators in CSS complexes form a
multiplicative system (abstract category) ~ of ·operators ~ such that
that. certain values of q (depending on ~,,..), ~ o q is defined and is
(for some p) a p-simplex of a CSS complex K for any q-simplex
4
Definition ~ = (4} is a multiplicative system generated by objects
l,o1,si(i 2: o), subject to the relations
Le't 4>n ,p be the set of ~ such that ~ CJn is defined and
*
~ CJ e B for any er e B • Let 4> be an anti-isomorphic copy of
n P n n
~ , and let ~* 9orrespond to ~ in the anti-isom9rphism (so that
4
( 4'1r) * = 'Ir* * ) . Also let ~* be the image of 4> in the
n,p n,p
anti-isomorphism.
Suppose M = {lf"} is a collection of CSS complexes "J:ll, n ~ O,
'!f'
(with usually consisting of one vertex and its degeneracies), and
suppose that 4>* is made to operate on M as a collection of CSS maps
MP lf".
in such a way that if ~ * e 4>* , then ~ *: ~ (For example,
. n,p
o; :
M°-l --.. lf" for each n 2:, i). Then define the star-functor M* by
Definition M*B is a CSS complex whose q-si•mplices are classes of pairs
I .
subject to the identifications
(2.21)
The face and degener?'cy operations in M*:S. are given by
(2.22)
5
If :fz B -+ c, then M*f: M*B -+ M* C is given by
·[Remark. (2.22),(2.23) are consistent with (2.21) since '* is a
CBS map, for any «j>* e ~ *.]
.(The realization functor can be similarly definedJ If' is
'*
there a geometric n-simplex, ' a simplicial map, (2.31) a set of
identifications),
If g: M -+ M is a collection of CSS maps gn: If' -+ Mn which
commute with the maps of 4>*(~* gP = gn q* if q* e 4>* )·, there
n, P
is a natural transformation gi M* -+ M* , such that
(2.24)
Examples of star functors. Conical tunctorz If' = join of the
. n
st~d n-simplex A with a fixed vertexJ suspension functor:
1'1 ..
Join,_·. of D.n with two fixed verticesJ baryc.entric division
functors defined below.
Let An be the (abstract) n-simplex (K [n] in [2] ) with
vertices o, .. _.,n. The order-preserviflg inclusion [o, ••• ,n-l]c[o, ••• ,nl
I
(image omitting 1), and the order-preserving projection
[Q, ••• ,n+l]-+[O, ••• ,n] (i,i+l-+ i) induce, respectively, simplicial
maps
6
* n n
Tb.is induces an operation of <l>, on ~ = (~ } • Let Sd~ = (N ~ )
(N = nerve functor), and let <I>* operate on Sd~ by
(~* € <i>* ) •
n,p
W
Similarly Sdr ~ can be defined for any r, as , ( ~) with
operators 1•~. .r ~'I' *.
(2.3) Definition SdrB = Sdr~ * B ; Sdrf = Sdr b. * f; r :::_ 0
Here Sd0 b. = b. •
Lemma. Tb.ere are natural isomorphisms B ~ Sd0B, Sdr+lB~ Sd(SdrB).
n
Proof. In the first case, let e e~ be the nondegenerate n-simplex.
n
n ·*
Tb.en any, -rqe.6; can be expressed uniquely as <t e for some p and
p
some ~'* e 11> n* ,p • Hence ( -r q , an ) = (~ * e p , an ) == (ep'~ an) e Sd0B, and
the map (e ,~a ) ~ ~ a is the desired isomorphism. In the
p n n
second case, notice that Nb.n = Sd b. n, and that M1 *M"*· = M* if
.Mn = M1 * M"n, f or any s t ar fun c t ors M1 *, M1 1 * . Identify
B, Sd0B; and Sd(SdrB), Sdr+lB by these isomorphisms.
The maps II.n : NLSn ~ ~n ( 2.1) ( such that 11.n(a~ =last vertex
of ag and, being a simplicial map, is uniquely described by this)
induce f..: Sdb. ~ b. and so ((2.34)) inducE:
(2.4) ":al SdB ~ B, -f.. : Sd -~ Sd0
since SdB = Sdb.*B, B = b.*B = Sd0 B.
7
3. Sd as a division functor.
Theorem 3 Sd is a division functor
The transformation ~: Sd -+ l is defined by (2.4), and is clearly
naturaIl . The definition of aB: jSdBI -+ IBI requires ~little
care, since the obvious homeomorphism jSd.6.n I -+ j.6.n I is not com
patible with identification maps jSds~ I, Is~ j.
In each face
arbitrary. Cover j.6.nl by a simplicial. complex isomorphic·to
Sd.6n (=N.6.n) by starring each cell in order of
increasing dimension q. [That is to say, when all cells of dimension
less than q have been triangulated, join each simplex on the boundary
of ei to A~, t = 0,1, .•. (~) ]. Then there is a natural barycentric
map t3 z jSd .6.n I ~ lt~PI defined by means of this triangulation, in
I p
which (ei) I ~ At.
For each on B,n, select (Ai) as follows. If the face a t of
E
p,
on corresponding to e~ (i.e. ~ en = e~, ~ on = op, t for some ~
E <I> )
is non-degenerate, then A~ is the barycentre (=centroid). If for
some 4, a = 4 a , then ~*: e~ '"4 eq is a simplicial
p, t p,u u
eq
map. For each vertex Vi of u' let Gi be the,centroid of the
vertices of e~ in ~*-1(Vi), and let AP be the centroid of the
t
Gi's (i= 0,1, ... ,q). This set of choices defines a triangulation of
IAnl
•and so a map
L::.
These maps, for all simplices of B, are easily shown to have the property
8
If .for some 4€ !IJn. ,.. m ' am = ~ on' then the diagram
~(on)
I
jsd Ll0 j :;:.. jAn
. l.
~*I 11~*1
IBd
t3( om)
I
jsdifll Ami
Commutes.
(It suffices to consider ~ = s1, ~ = oi only).
l n
The (unique) CSS map [a ] : A --+ B which maps e on on
n n
has a realization
while Sd [an] : SdL'P --+ Sd B similarly defines
It follows from the definition of jBI, !SdBI as identification spaces
·that the map
is single valued and so continuous ( [ 8] ) , l -1 and so a homeo-
IB'I j I
morphism ( is Hausdorff, g( a ) Sd An is compact). Al.so, under
. n
~
the hypotheses of (3~1), g(om) 0 jsa~*I ""g(on); f(am)0 lt*I = f(a0 H
I /
and hence by (3.l), (i B, defined piecewise over jSdB is a
9
a
I
homeomorphism •. (CiB is continuous, since Sd B I · is CW complex,
a:
.an~has a continuous inverse similarly constructed).
Now (i B 'ct' r~ B I j for, with the triangulation Of IA n I under
~(on)' define ht(on)i jAnj -+ jAnl such that h (on) = 1, ht(on)IA~
0 ...
.
.
is a linear map of A~X I on the segment joining A~ to the last
vertex of ei, and ht(on) is a linear extension of this. These
IB I IB I·
homotopies define Ht: -4 by
= f( o ) o ht( a ) o f( o )-1•
n n n
a a I .
Tb.en it is clear that Ho B = B ; ~ aB = I~ B This completes
the proof of Theorem 3,
4. Regular Complexes
I
Definition A CSS complex is regular if each non-degenerate on
has a vertex such that no face of on this
conta~ning
vertex is degenerate.
Definition A CW complex is regular if the closure of each n-cell
is an n-element,·and if e1,e2 are two cells, then
e3
for some cell e3' ~ne2 = (or=~).
Theorem 4. The realization of a regular CSS complex is a regular CW
complex, and so can be ,trianguiated by starring each cell
in order of increasing dimension.
1,;,J
