Geometrical Theory on Combinatorial Manifolds

2006·0.3 MB·

by  Linfan Mao

Geometrical Theory on C¸ ombinatorial Manifolds
6 Linfan¸ Mao
0
0 (ChineseAcademyofMathematicsandS¸ystemScience, Beijing100080, P.R.China)
2
E-mail: maoli¸[email protected]

c
e
D Abstract: Foranintegerm ≥ 1,acombinatorialmanifoldM isdefinedtobe
6 ageometricalobjectM suchthatfor∀p∈ M,thereisalocalfchart(Up,ϕp)en-
] 2 ∅a,blwehϕepre: UBpni→j iBs nain1fSniBj-nbia2lSl f·o·r·SintBegniesr(ps)f1wi≤thjB≤ni1sT(pB)n≤i2 Tm·.··TToBponlios(gpi)ca6=l
M
and differential structures such as those of d-pathwise connected, homotopy
G classes, fundamental d-groups in topology and tangent vector fields, tensor
. fields, connections, Minkowski norms in differential geometry on these finitely
h
t combinatorial manifolds are introduced. Some classical results are generalized
a
m to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed
[ and geometrical inclusions in Smarandache geometries for various geometries

are also presented by the geometrical theory on finitely combinatorial mani-
1
v folds in this paper.
0
6 Key Words: manifold, finitely combinatorial manifold, topological struc-
7
ture, differential structure, combinatorially Riemannian geometry, combinato-
2
1 rially Finsler geometry, Euler-Poincare characteristic.
6
0 AMS(2000): 51M15, 53B15, 53B40, 57N16
/
h
t §1. Introduction
a
m
As a model of spacetimes in physics, various geometries such as those of Euclid,
:
v RiemannianandFinsler geometriesareestablished bymathematicians. Today, more
i
X and more evidences have shown that our spacetime is not homogenous. Thereby
r models established on classical geometries are only unilateral. Then are there some
a
kinds of overall geometries for spacetimes in physics? The answer is YES. Those
are just Smarandache geometries established in last century but attract more one’s
attention now. According to thesummary in [4], they areformally defined following.
Definition 1.1([4],[17]) A Smarandache geometry is a geometry which has at least
one Smarandachely denied axiom(1969), i.e., an axiom behaves in at least two dif-
ferent ways within the same space, i.e., validated and invalided, or only invalided
but in multiple distinct ways.
A Smarandache n-manifold is a n-manifold that support a Smarandache geome-
try.
For verifying the existence of Smarandache geometries, Kuciuk and Antholy
gave a popular and easily understanding example on an Euclid plane in [4]. In
[3], Iseri firstly presented a systematic construction for Smarandache geometries
1
by equilateral triangular disks on Euclid planes, which are really Smarandache 2-
dimensionalgeometries(seealso[5]). Inreferences [6],[7]and[13],particularlyin[7],
a general constructing way for Smarandache 2-dimensional geometries on maps on
surfaces, called map geometries was introduced, which generalized the construction
of Iseri. For the case of dimensional number≥ 3, these pseudo-manifold geometries
are proposed, which are approved to be Smarandache geometries and containing
these Finsler and K¨ahler geometries as sub-geometries in [12].
In fact, by the Definition 1.1 a general but more natural way for constructing
Smarandache geometries should be seeking for them on a union set of spaces with an
axiom validated in one space but invalided in another, or invalided in a space in one
way and another space in a different way. These unions are so called Smarandache
multi-spaces. This is the motivation for this paper. Notice that in [8], these multi-
metric spaces have been introduced, which enables us to constructing Smarandache
geometries on multi-metric spaces, particularly, on multi-metric spaces with a same
metric.
Definition 1.2 A multi-metric space A is a union of spaces A ,A ,···,A for an
1 2 m
integer k ≥ 2 such that each A is a space with metric ρ for ∀i,1 ≤ i ≤ m.
i i
e
Now for any integer n, these n-manifolds Mn are the main objects in modern
geometry and mechanics, which are locally euclidean spaces Rn satisfying the T
2
separation axiom in fact, i.e., for ∀p,q ∈ Mn, there are local charts (U ,ϕ ) and
p p
(U ,ϕ ) such that U U = ∅ and ϕ : U → Bn, ϕ : U → Bn, where
q q p q p p q q
T
Bn = {(x ,x ,···,x )|x2 +x2 +···+x2 < 1}.
1 2 n 1 2 n
is an open ball.
These manifolds are locally euclidean spaces. In fact, they are also homogenous
spaces. But the world is not homogenous. Whence, a more important thing is
considering these combinations of different dimensions, i.e., combinatorial manifolds
defined following and finding their good behaviors for mathematical sciences besides
just to research these manifolds. Two examples for these combinations of manifolds
with different dimensions in R3 are shown in Fig.1.1, in where, (a) represents a
combinationofa3-manifold,atorusand1-manifold,and(b)atoruswith4bouquets
of 1-manifolds.
Fig¸.1.1
2
For an integer s ≥ 1, let n ,n ,···,n be an integer sequence with 0 < n <
1 2 s 1
s
n < ··· < n . Choose s open unit balls Bn1,Bn2,···,Bns, where Bni 6= ∅ in
2 s 1 2 s i
i=1
T
Rn1+2+···ns. Then a unit open combinatorial ball of degree s is a union
s
B(n ,n ,···,n ) = Bni.
1 2 s i
i[=1
e
Definition 1.3 For a given integer sequence n ,n ,···,n ,m ≥ 1 with 0 < n <
1 2 m 1
n < ··· < n , a combinatorial manifold M is a Hausdorff space such that for
2 s
any point p ∈ M, there is a local chart (U ,ϕ ) of p, i.e., an open neighborhood
p p
f
U of p in M and a homoeomorphism ϕ : U → B(n (p),n (p),···,n (p)) with
p f p p 1 2 s(p)
{n (p),n (p),···,n (p)} ⊆ {n ,n ,···,n } and {n (p),n (p),···,n (p)} =
1 2 f s(p) 1 2 m e 1 2 s(p)
p∈SM
f
{n ,n ,···,n }, denoted by M(n ,n ,···,n ) or M on the context and
1 2 m 1 2 m
f f
A = {(U ,ϕ )|p ∈ M(n ,n ,···,n ))}
p p 1 2 m
an atlas on M(n ,n ,··e·,n ). The maximfum value of s(p) and the dimension s(p)
1 2 m
s(p)
of Bni arfe called the dimension and the intersectional dimensional of M(n b,n ,
i 1 2
i=1
T
···,n ) at the point p, respectively. f
m
A combinatorial manifold M is called finite if it is just combined by finite man-
ifolds.
s f
Notice that Bni 6= ∅ by the definition of unit combinatorial balls of degree
i
i=1
T
s. Thereby, for ∀p ∈ M(n ,n ,···,n ), either it has a neighborhood U with ϕ :
1 2 s p p
U → Rς, ς ∈ {n ,n ,···,n } or a combinatorial ball B(τ ,τ ,···,τ ) with ϕ :
p 1 2f s 1 2 l p
U → B(τ ,τ ,···,τ ), l ≤ s and {τ ,τ ,···,τ } ⊆ {n ,n ,···,n } hold.
p 1 2 l 1 2 l 1 2 e s
The main purpose of this paper is to characterize these finitely combinatorial
e
manifolds, such as those of topological behaviors and differential structures on them
by a combinatorial method. For these objectives, topological and differential struc-
turessuchasthoseofd-pathwiseconnected, homotopyclasses, fundamentald-groups
in topology and tangent vector fields, tensor fields, connections, Minkowski norms in
differential geometry on these combinatorial manifolds are introduced. Some results
in classical differential geometry are generalized to finitely combinatorial manifolds.
As an important invariant, Euler-Poincare characteristic is discussed and geomet-
rical inclusions in Smarandache geometries for various existent geometries are also
presented by the geometrical theory on finitely combinatorial manifolds in this pa-
per.
For terminologies and notations not mentioned in this section, we follow [1]−[2]
for differential geometry, [5],[7] for graphs and [14],[18] for topology.
3
§2. Topological structures on combinatorial manifolds
By a topological view, we introduce topological structures and characterize these
finitely combinatorial manifolds in this section.
2.1. Pathwise connectedness
On the first, we define d-dimensional pathwise connectedness in a finitely combina-
torial manifold for an integer d,d ≥ 1, which is a natural generalization of pathwise
connectedness in a topological space.
Definition 2.1 For two points p,q in a finitely combinatorial manifold M(n ,n ,
1 2
···,n ), if there is a sequence B ,B ,···,B of d-dimensional open balls with two
m 1 2 s
f
conditions following hold.
(1) B ⊂ M(n ,n ,···,n ) for any integer i,1 ≤ i ≤ s and p ∈ B , q ∈ B ;
i 1 2 m 1 s
(2) The dimensional number dim(B B ) ≥ d for ∀i,1 ≤ i ≤ s−1.
i i+1
f
T
Then points p,q are called d-dimensional connected in M(n ,n ,···,n ) and the se-
1 2 m
quence B ,B ,···,B a d-dimensional path connecting p and q, denoted by Pd(p,q).
1 2 e
f
If each pair p,q of points in the finitely combinatorial manifold M(n ,n ,···,n )
1 2 m
is d-dimensional connected, then M(n ,n ,···,n ) is called d-pathwise connected
1 2 m f
and say its connectivity≥ d.
f
Not loss of generality, we consider only finitely combinatorial manifolds with
a connectivity≥ 1 in this paper. Let M(n ,n ,···,n ) be a finitely combinatorial
1 2 m
manifoldandd,d ≥ 1aninteger. WeconstructalabelledgraphGd[M(n ,n ,···,n )]
f 1 2 m
by
f
V(Gd[M(n ,n ,···,n )]) = V V ,
1 2 m 1 2
[
where V = {n − manifoldsfMni in M(n ,n ,···,n )|1 ≤ i ≤ m} and V =
1 i 1 2 m 2
{isolated intersection points OMni,Mnj fofMni,Mnj in M(n1,n2,···,nm) for 1 ≤
i,j ≤ m}. Label n for each n -manifold in V and 0 for each vertex in V and
i i 1 f 2
E(Gd[M(n ,n ,···,n )]) = E E ,
1 2 m 1 2
[
where E = {(Mni,Mnj)|dfim(Mni Mnj) ≥ d,1 ≤ i,j ≤ m} and E =
1 2
{(OMni,Mnj,Mni),(OMni,Mnj,Mnj)|MnTi tangent Mnj at the point OMni,Mnj for 1 ≤
i,j ≤ m}.
4
Fig¸.2.1
For example, these correspondent labelled graphs gotten from finitely combina-
torialmanifolds in Fig.1.1 are shown in Fig.2.1, in where d = 1 for (a) and (b), d = 2
for (c) and (d). By this construction, properties following can be easily gotten.
Theorem 2.1 Let Gd[M(n ,n ,···,n )] be a labelled graph of a finitely combina-
1 2 m
torial manifold M(n ,n ,···,n ). Then
1 f2 m
(1) Gd[M(n ,n ,···,n )] is connected only if d ≤ n .
1f2 m 1
(2)there exists an integer d,d ≤ n such that Gd[M(n ,n ,···,n )] is connected.
f 1 1 2 m
Proof By definition, there is an edge (Mni,Mnfj) in Gd[M(n ,n ,···,n )] for
1 2 m
1 ≤ i,j ≤ m if and only if there is a d-dimensional path Pd(p,q) connecting two
f
points p ∈ Mni and q ∈ Mnj. Notice that
(Pd(p,q)\Mni) ⊆ Mnj and (Pd(p,q)\Mnj) ⊆ Mni.
Whence,
d ≤ min{n ,n }. (2.1)
i j
Now if Gd[M(n ,n ,···,n )] is connected, then there is a d-path P(Mni,Mnj)
1 2 m
connecting vertices Mni and Mnj for ∀Mni,Mnj ∈ V(Gd[M(n ,n ,···,n )]). Not
f 1 2 m
loss of generality, assume
f
P(Mni,Mnj) = MniMs1Ms2 ···Mst−1Mnj.
Then we get that
d ≤ min{n ,s ,s ,···,s ,n } (2.2)
i 1 2 t−1 j
by (2.1). However, according to Definition 1.4 we know that
5
{n (p),n (p),···,n (p)} = {n ,n ,···,n }. (2.3)
1 2 s(p) 1 2 m
[
p∈M
f
Therefore, we get that
d ≤ min( {n (p),n (p),···,n (p)}) = min{n ,n ,···,n } = n
1 2 s(p) 1 2 m 1
[
p∈M
f
by combining (2.3) with (2.3). Notice that points labelled with 0 and 1 are always
connected by a path. We get the conclusion (1).
For the conclusion (2), notice that any finitely combinatorial manifold is al-
ways pathwise 1-connected by definition. Accordingly, G1[M(n ,n ,···,n )] is con-
1 2 m
nected. Thereby, there at least one integer, for instance d = 1 enabling Gd[M(n ,n ,
f 1 2
···,n )] to be connected. This completes the proof. ♮
m
f
According to Theorem 2.1, we get immediately two corollaries following.
Corollary 2.1 For a given finitely combinatorial manifold M, all connected graphs
Gd[M] are isomorphic if d ≤ n , denoted by G[M].
1 f
Corfollary 2.2 If there are k 1-manifolds intfersect at one point p in a finitely
combinatorial manifold M, then there is an induced subgraph Kk+1 in G[M].
Now we define an edfge set Ed(M) in G[M] by f
f f
Ed(M) = E(Gd[M])\E(Gd+1[M]).
Then we get a graphical recfursion equaftion for graphfs of a finitely combinatorial
manifold M as a by-product.
f
Theorem 2.2 Let M be a finitely combinatorial manifold. Then for any integer
d,d ≥ 1, there is a recursion equation
f
Gd+1[M] = Gd[M]−Ed(M)
for graphs of M. f f f
Proof It cfan be obtained immediately by definition. ♮
For a given integer sequence 1 ≤ n < n < ··· < n ,m ≥ 1, denote by
1 2 m
Hd(n ,n ,···,n ) all these finitely combinatorial manifolds M(n ,n ,···,n ) with
1 2 m 1 2 m
connectivity≥ d, where d ≤ n and G(n ,n ,···,n ) all these connected graphs
1 1 2 m f
G[n ,n ,···,n ] with vertex labels 0,n ,n ,···,n and conditions following hold.
1 2 m 1 2 m
(1) The induced subgraph by vertices labelled with 1 in G is a union of complete
graphs;
(2) All vertices labelled with 0 can only be adjacent to vertices labelled with 1.
6
Then we know a relation between sets Hd(n ,n ,···,n ) and G(n ,n ,···,n ).
1 2 m 1 2 m
Theorem 2.3 Let 1 ≤ n < n < ··· < n ,m ≥ 1 be a given integer sequence. Then
1 2 m
every finitely combinatorial manifold M ∈ Hd(n ,n ,···,n ) defines a labelled con-
1 2 m
nected graph G[n ,n ,···,n ] ∈ G(n ,n ,···,n ). Conversely, every labelled con-
1 2 m 1f 2 m
nected graph G[n ,n ,···,n ] ∈ G(n ,n ,···,n ) defines a finitely combinatorial
1 2 m 1 2 m
manifold M ∈ Hd(n ,n , ···,n ) for any integer 1 ≤ d ≤ n .
1 2 m 1
Proof fFor∀M ∈ Hd(n ,n ,···,n ), thereisalabelled graphG[n ,n ,···,n ] ∈
1 2 m 1 2 m
G(n ,n ,···,n ) correspondent to M is already verified by Theorem 2.1. For
1 2 mf
completing the proof, we only need to construct a finitely combinatorial manifold
f
M ∈ Hd(n ,n ,···,n ) for ∀G[n ,n ,···,n ] ∈ G(n ,n ,···,n ). Denoted by
1 2 m 1 2 m 1 2 m
l(u) = s if the label of a vertex u ∈ V(G[n ,n ,···,n ]) is s. The construction is
1 2 m
f
carried out by the following programming.
STEP 1. Choose |G[n ,n ,···,n ]|−|V | manifolds correspondent to each vertex u
1 2 m 0
withadimensionaln ifl(u) = n ,whereV = {u|u ∈ V(G[n ,n ,···,n ])andl(u) =
i i 0 1 2 m
0}. Denoted by V all these vertices in G[n ,n ,···,n ] with label≥ 1.
≥1 1 2 m
STEP 2. For∀u ∈ V with l(u ) = n , if its neighborhood set N (u )
1 ≥1 1 i1 G[n1,n2,···,nm] 1
V = {v1,v2,···,vs(u1)} with l(v1) = n , l(v2) = n , ···, l(vs(u1)) = n , theTn
≥1 1 1 1 1 11 1 12 1 1s(u1)
let the manifold correspondent to the vertex u with an intersection dimension≥ d
1
with manifolds correspondent to vertices v1,v2,···,vs(u1) and define a vertex set
1 1 1
∆ = {u }.
1 1
STEP 3. If the vertex set ∆ = {u ,u ,···,u } ⊆ V has been defined and V \
l 1 2 l ≥1 ≥1
∆ 6= ∅, let u ∈ V \∆ with a label n . Assume
l l+1 ≥1 l il+1
(N (u ) V )\∆ = {v1 ,v2 ,···,vs(ul+1)}
G[n1,n2,···,nm] l+1 ≥1 l l+1 l+1 l+1
\
with l(v1 ) = n , l(v2 ) = n , ···,l(vs(ul+1)) = n . Then let the
l+1 l+1,1 l+1 l+1,2 l+1 l+1,s(ul+1)
manifold correspondent to the vertex u with an intersection dimension≥ d with
l+1
manifolds correspondent to these vertices v1 ,v2 ,···,vs(ul+1) and define a vertex
l+1 l+1 l+1
set ∆ = ∆ {u }.
l+1 l l+1
S
STEP 4. Repeat steps 2 and 3 until a vertex set ∆ = V has been constructed.
t ≥1
This construction is ended if there are no vertices w ∈ V(G) with l(w) = 0, i.e.,
V = V(G). Otherwise, go to the next step.
≥1
STEP 5. For ∀w ∈ V(G[n ,n ,···,n ])\V , assume N (w) = {w ,w ,
1 2 m ≥1 G[n1,n2,···,nm] 1 2
···,w }. Let all these manifolds correspondent to vertices w ,w ,···,w intersects
e 1 2 e
at one point simultaneously and define a vertex set ∆∗ = ∆ {w}.
t+1 t
S
STEP6. RepeatSTEP5forverticesinV(G[n ,n ,···,n ])\V . Thisconstruction
1 2 m ≥1
is finally ended until a vertex set ∆∗ = V(G[n ,n ,···,n ]) has been constructed.
t+h 1 2 m
As soon as the vertex set ∆∗ has been constructed, we get a finitely combi-
t+h
7
natorial manifold M. It can be easily verified that M ∈ Hd(n ,n ,···,n ) by our
1 2 m
construction way. ♮
f f
2.2 Combinatorial equivalence
For a finitely combinatorial manifold M in Hd(n ,n ,···,n ), denoted by G[M(n ,
1 2 m 1
n ,···,n )] and G[M] the correspondent labelled graph in G(n ,n ,···,n ) and
2 m f 1 2 mf
the graph deleted labels on G[M(n ,n ,···,n )], C(n ) all these vertices with a
f 1 2 m i
label n for 1 ≤ i ≤ m, respectively.
i
f
Definition 2.2 Two finitely combinatorial manifolds M (n ,n ,···,n ),M (k ,k ,
1 1 2 m 2 1 2
···,k ) are called equivalent if these correspondent labelled graphs
l
f f
G[M (n ,n ,···,n )] ∼= G[M (k ,k ,···,k )].
1 1 2 m 2 1 2 l
f f
Notice that if M (n ,n ,···,n ),M (k ,k ,···,k ) are equivalent, then we can
1 1 2 m 2 1 2 l
get that {n ,n ,···,n } = {k ,k ,···,k } and G[M ] ∼= G[M ]. Reversing this idea
1 2 f m 1 2 fl 1 2
enables us classifying finitely combinatorial manifolds in Hd(n ,n ,···,n ) by the
1 2 m
f f
action of automorphism groups of these correspondent graphs without labels.
Definition 2.3 A labelled connected graph G[M(n ,n ,···,n )] is combinatorial
1 2 m
unique if all these correspondent finitely combinatorial manifolds M(n ,n ,···,n )
f 1 2 m
are equivalent.
f
A labelled graph G[n ,n ,···,n ] is called class-transitive if the automorphism
1 2 m
group AutG is transitive on {C(n ),1 ≤ i ≤ m}. We find a characteristic for
i
combinatorially unique graphs.
Theorem 2.4 A labelled connectedgraph G[n ,n ,···,n ] is combinatoriallyunique
1 2 m
if and only if it is class-transitive.
Proof For two integers i,j,1 ≤ i,j ≤ m, re-label vertices in C(n ) by n and
i j
vertices in C(n ) by n in G[n ,n ,···,n ]. Then we get a new labelled graph
j i 1 2 m
G′[n ,n ,···,n ] in G[n ,n ,···,n ]. According to Theorem 2.3, we can get two
1 2 m 1 2 m
finitelycombinatorialmanifoldsM (n ,n ,···,n )andM (k ,k ,···,k )correspon-
1 1 2 m 2 1 2 l
dent to G[n ,n ,···,n ] and G′[n ,n ,···,n ].
1 2 m f1 2 m f
Now if G[n ,n ,···,n ] is combinatorially unique, we know M (n ,n ,···,n )
1 2 m 1 1 2 m
is equivalent to M (k ,k ,···,k ), i.e., there is an automorphism θ ∈ AutG such
2 1 2 l f
that Cθ(n ) = C(n ) for ∀i,j,1 ≤ i,j ≤ m.
i j
f
Ontheotherhand,ifG[n ,n ,···,n ]isclass-transitive, thenforintegersi,j,1 ≤
1 2 m
i,j ≤ m, there is an automorphism τ ∈ AutG such that Cτ(n ) = C(n ). Whence,
i j
for any re-labelled graph G′[n ,n ,···,n ], we find that
1 2 m
G[n ,n ,···,n ] ∼= G′[n ,n ,···,n ],
1 2 m 1 2 m
8
which implies that these finitely combinatorial manifolds correspondent to G[n ,n ,
1 2
···,n ] and G′[n ,n ,···,n ] are combinatorially equivalent, i.e., G[n ,n ,···,n ]
m 1 2 m 1 2 m
is combinatorially unique. ♮
Now assume that for parameters t ,t ,···,t , we have known an enufunction
i1 i2 isi
C [x ,x ,···] = n (t ,t ,···,t )xti1xti2 ···xtis
Mni i1 i2 i i1 i2 is i1 i2 is
ti1,tXi2,···,tis
for n -manifolds, where n (t ,t ,···,t ) denotes the number of non-homeomorphic
i i i1 i2 is
n -manifolds with parameters t ,t ,···,t . For instance the enufunction for com-
i i1 i2 is
pact 2-manifolds with parameter genera is
C [x](2) = 1+ 2xp.
M
f
Xp≥1
Consider the action of AutG[n ,n ,···,n ] on G[n ,n ,···,n ]. If the number of
1 2 m 1 2 m
orbits of the automorphism group AutG[n ,n ,···,n ] action on {C(n ),1 ≤ i ≤
1 2 m i
m} is π , then we can only get π ! non-equivalent combinatorial manifolds corre-
0 0
spondent to the labelled graph G[n ,n ,···,n ] similar to Theorem 2.4. Calcula-
1 2 m
tion shows that there are l! orbits action by its automorphism group for a complete
(s + s + ···+ s )-partite graph K(ks1,ks2,···,ksl), where ksi denotes that there
1 2 l 1 2 l i
are s partite sets of order k in this graph for any integer i,1 ≤ i ≤ l, particularly,
i i
for K(n ,n ,···,n ) with n 6= n for i,j,1 ≤ i,j ≤ m, the number of orbits action
1 2 m i j
by its automorphism group is m!. Summarizing all these discussions, we get an enu-
function for these finitely combinatorial manifolds M(n ,n ,···,n ) correspondent
1 2 m
to a labelled graph G[n ,n ,···,n ] in G(n ,n ,···,n ) with each label≥ 1.
1 2 m 1 2 m
f
Theorem2.5 Let G[n ,n ,···,n ]be alabelledgraphin G(n ,n ,···,n ) with each
1 2 m 1 2 m
label≥ 1. For an integer i,1 ≤ i ≤ m, let the enufunction of non-homeomorphic n -
i
manifolds with given parameters t ,t ,···, be C [x ,x ,···] and π the number
1 2 Mni i1 i2 0
of orbits of the automorphism group AutG[n ,n ,···,n ] action on {C(n ),1 ≤ i ≤
1 2 m i
m}, then the enufunction of combinatorial manifolds M(n ,n ,···,n ) correspon-
1 2 m
dent to a labelled graph G[n ,n ,···,n ] is
1 2 m f
m
C (x) = π ! C [x ,x ,···],
M 0 Mni i1 i2
f
Yi=1
particularly, if G[n ,n ,···,n ] = K(ks1,ks2,···,ksm) such that the number of par-
1 2 m 1 2 m
tite sets labelled with n is s for any integer i,1 ≤ i ≤ m, then the enufunction
i i
correspondent to K(ks1,ks2,···,ksm) is
1 2 m
m
C (x) = m! C [x ,x ,···]
M Mni i1 i2
f
Yi=1
and the enufunction correspondent to a complete graph K is
m
9
m
C (x) = C [x ,x ,···].
M Mni i1 i2
f
Yi=1
Proof Notice that the number of non-equivalent finitely combinatorial manifolds
correspondent to G[n ,n ,···,n ] is
1 2 m
m
π n (t ,t ,···,t )
0 i i1 i2 is
Yi=1
for parameters t ,t ,···,t ,1 ≤ i ≤ m by the product principle of enumeration.
i1 i2 is
Whence, the enufunction of combinatorial manifolds M(n ,n ,···,n ) correspon-
1 2 m
dent to a labelled graph G[n ,n ,···,n ] is
1 2 m f
m m
C (x) = (π n (t ,t ,···,t )) xti1xti2 ···xtis
M 0 i i1 i2 is i1 i2 is
f
ti1,tXi2,···,tis Yi=1 Yi=1
m
= π ! C [x ,x ,···]. ♮
0 Mni i1 i2
Yi=1
2.3 Homotopy classes
Denote by f ≃ g two homotopic mappings f and g. Following the same pattern of
homotopic spaces, we define homotopically combinatorial manifolds in the next.
Definition2.4 Two finitelycombinatorialmanifoldsM(k ,k ,···,k ) andM(n ,n ,
1 2 l 1 2
···,n ) are said to be homotopic if there exist continuous maps
m
f f
f : M(k ,k ,···,k ) → M(n ,n ,···,n ),
1 2 l 1 2 m
g : M(n ,n ,···,n ) → M(k ,k ,···,k )
f 1 2 m f 1 2 l
such that gf ≃identity: M(k ,k ,···,k ) → M(k ,k ,···,k ) and fg ≃identity:
f f1 2 l 1 2 l
M(n ,n ,···,n ) → M(n ,n ,···,n ).
1 2 m f1 2 m f
f For equivalent homfotopically combinatorial manifolds, we know the following
result under these correspondent manifolds being homotopic. For this objective, we
need an important lemma in algebraic topology.
Lemma 2.1(Gluing Lemma, [16]) Assume that a space X is a finite union of closed
n
subsets: X = X . If for some space Y, there are continuous maps f : X → Y
i i i
iS=1
that agree on overlaps, i.e., f | = f | for all i,j, then there exists a
i XiTXj j XiTXj
unique continuous f : X → Y with f| = f for all i.
Xi i
Theorem 2.6 Let M(n ,n ,···,n ) and M(k ,k ,···,k ) be finitely combinato-
1 2 m 1 2 l
rial manifolds with an equivalence ̟ : G[M(n ,n ,···,n )] → G[M(k ,k ,···,k )].
f f1 2 m 1 2 l
f f
10