topology in mathematics a molecular structure of...
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Title Topology in Mathematics A MOLECULAR STRUCTURE OFMANIFOLDS
Author(s) Nagamine, Yasunobu
Citation沖大論叢.人文科学・社会科学・自然科学・英語英文学 =OKIDAI REVIEW OF ENGLISH AND GENERALEDUCATION, 1(1): 21-36
Issue Date 1975-03-31
URL http://hdl.handle.net/20.500.12001/10603
Rights 沖縄大学英語科・教養部
Topology in Mathematics
A MoLECULAR STRUCTURE oF MANIFOLDS
Yasunobu. Nagamine
Preface
Previously, the theory of topology has been practic
ed ~y Prof. P. Alexandvoff, [Bi 14], I. S. Pontrjagin
[8]. Recently it has been developped by Professors Whi
tehead [4], Steenrod [6], R: Them, J. Milnor [5]. But, I think that homology, homotopy and groups are
an electronic theory of manifolds. Hence, we can not
exactly express all character of manifolds.
Recently, I have tried to express a weight of mani
folds and complexes. I suppose that there exists an
aspect·, a form and a charactuer in it. There exists a
molecular weight of it.
If we put a vertex on the other face and we try
similarly many times, at last, it becomes to a minimam
charactristic complex. And we shall call it "the mole
cular complex." ·There exists a molecular weight:
·* {F=!.-2x(P)}
Also, it is invariant for the topological maps. I think
that it will be applicational in all categories.
-21-
§I A weight of Complex
In the space E~ we well know that the inner angle of
n-palygon is given by the following:
f=mr-2x(P)Tr
(xis the Euler's characteristic).
f is invariant as topological maps. Now, dividing F by
TI, we have the next:
F=n-2x(P)
we can define that F is a weight of a complex. As
[m]-rolling palygon, we can get a weight the next form:
F=/..-2m
In the space E; we shall represent the sum of a facal
angle of a palyhedron. Now on a vertex, we can draw a
sphere and the normal unit vector.
Similer to the Legendre's proof, we can prove
f=2/..TI-2x(P)TI
Dividing by TI, we ~et F=2/..-2x(P) or 2/..-4m.
(m is the degree of normal vector on P ).
Therefore we can have the next theovem:
(Theovem I) "In the space E; all complexes have a wei
ght of itself and it is given by the form:
F=2/..-4m
(/..: number of ventex, m: the degree of the
normal vector)"
-22-
[For example, the case of a palyhedron m=l, trus: m=O]
Next, in the space E; we shall consider a complex insc
ribing in a manifold. When we run over on the above
complex, there arises the complexity of this complex.
It is a weight of this complex.
Then, we can determine in the following: . 7rT
J =<. l- 2 X) F( -i-). a"T Deviding it by rc -r>.
we can unify in the next form:
F=(A.-2x)
Hence, we can have the next theorem:
(Theorem II) "In the space E; all complex have a weight
of itself.
It is given by the next:
F=A.-2x
(A~n+l)"
§II A Molecular Weight of Manifolds
In the space E; on the (n-1)-complex, if we shall
push a vertex (al) on the face (a2, a3··········an+l),
it's weight reduce 1 from F(A-2x)
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This moving act is one of the moving groups, and it is
the lie group. We can easily prove that 'Vi deforms a
group. (§III)
If we continue to push a1 on a 2 ; a 2 on a 3 ; ••••••• ,
at last we arrive to the minimum character complex.
The number of a weight reduces little by little.
It unifies a molecular weight which is simmilar to a
molecular of all matters. We shall call it "amolecular
weight of manifolds". Hence we can assume the next
theorem:
(Theoren III) "All manifolds in the space E"have a mol
ecular weight of itself. It is expressed by the form:
* * F=A-2x(P)".
(A contains a dimmension, x(P) co11tains a form and a
character of manifold.)
§III Lie Group and Lie Ring of a Floating Group
('V; ' /1;)
In the space V(n)', ~e have already seen that the
inner points are expressed by the barycentric coordena
te. Where a simplex M(n) is denoted the next form:
P=lliVI+llaV2+ •••• +lln+l v n+I
Moreover, a complex in the space V(n) is expressed by
the barycentric coordinate in the fallowing:
P=ll1Vl+ll2V2+ ••••
If we will fix A-vectors (v1 v 2 ••••• v,) (A>n+l) in
-24-
the space V(n), it is expressed by a matrix in the next
form:
•• •• 0 ]'. (I:JII=l) . .. }lA
All points in A.-complex are unified by a variation of
~i • The above matrix space is represented by a A.-dime
nsional vector space.
Namelly A.-complexis equivalent ·to a matrix space
[ ~1,. •.•. ~i. ]( A.~n+l).
Next, we can define the folowing as addition of two
matrixes:
[~1· ·. ~). ]+[ v1 .•. v .. J=[~1+v1 •.• ,~. +v,].
As multiplication of two matrixes, moreover, we can
define in the following:
(JII) X (Vi) = (JIJVu JlzV2. ••• ···, J1 AvA).
(where : I:Jii=1, I:vi= 1, I:JIIvi < 1).
Now, we shall study a property of matrix. And we
can represent a exponention (eA) of matrix (A) which is
expressed in the next:
-25-
0
0 ··· ...
If we try to use determination of ( expA), we can
have the next calculation:
det ( expA) =e.., . e"' ...... e" 1
=exp(~l+ ....... +~
=e .
Hence, we have the next theorem:
(Theorem IV) det (expA) =e
By multiplication of (expA) and (expB), we can have the
next various farmula:
(Theorem V) i) det (expA) 2 =e 2
ii) det (expA)n=en
iii) tre (A) >tre A2 > ... >tre An> ....
Next, we shall study Lie group and Lie ring of the
floating groups (V; , ~; ).
We can unifty an infinitive minute map in the follow
ing:
-26-
dA at-
0
0
J.lA
(Note: ~~+~2+ ..... +~.=0)
Thus, ~; beccmes at least negative. If we use 'the
motion [V; (~;+0)], we eaasily see that ~;reduces little
by little.
The point (P) keeps a distance from V; and it comes
near the face (v2 v3·····Vm ):
vil. ,
When ~ moves in various values, P is floating in
whole poins of the complex (v 1v 2 •••• v. ). They are flo
ating groups.
Now, we can see a local structure of neighborhood (U)
-27-
in a complex (M). We can exactly denote it by lie ring
(exptX). Here, we call it "Lie ring of the floating
groups". And we can prove the next thorem (The continu
ous groups [10] L.S. Pontrijagin):
(Theorem VI) i) (A(t)+B(t)) '=A' (t)+B' (t)
ii) (A·B) '=A' ·B+A·B'
By the continuous group [10] we have the next theorem:
(Theorem VII) i) X,Y is finite
ii) (expX)· (expY)=ex+y +0( X Y )
iii) log{(expX)(expY)}=X+Y+O( X Y ).
When X or Y is a infinitive minute map, we have var
ious formulas the next:
(Theorem VIII) i) det (expX·expY)=l
ii) det (expX)n =1
Moreover, we have:
(Theorem IX) As a infinitive minute map (X,Y)
We should have the next limiting:
( X y )" Lim exp - · exp -....... n n
= exp (X + Y).
§IV A Sequence of Equvalence
If two manifolds is homomorphism, we can have the
next homological form:
-28-
* ~:H(Pr);;;H(Pr) But, it is not always realize the next form:
~:F(P);;,F(P)
In order to be equvalent, We need the next equalities:
X.=A.'
x=x'
Now, pushing many times, there arise the equvalent seq
uence in the form:
F( P r+ 1 )_]2".,... ....
VlY 'F(Pr) 'F(Pr-1) 0
I ... I I (A.-2x)
' (T+ 1) A- 2 X? (r
(p- 2 x) ~o ( r-'- 1 l
* In each dimensional space, F (weight) and homology is
invariant to topological maps (§:P P). Then, we shall
obtiain the next equvalence.
IJ:Fr(P};;Fr(P}
~:Fr(P};;Fr(P)
Here, we can have the next sequence of homologies and a
weights.
-29-
00 00
II: HOMOLOGY vJj~ v t l~ ~: A WEIGHT (p") (p")
F: A I"DLECULAR WE I GHT '0 V .], t ~ \ '0 V ~ 1 ~ \
~ FCP') \ ~ F(p') \ v I 1\!:::. \ I 1") v ~ 1 ~ \ \
* * .V I ' ' ,~ * ' ' V ~ F~~l ', ~ FC~) , ' v F~~l ,,' ~,'. 0 ' ~ t '\ ~ i ' 'H(p) c a > ',II~P) (i) a ~ ' HCP) ~
f,;jr+1 * f,; f,;\r * f,; f,;~r-1 . . _ a* a*
*Fe-) ,,-fJC~l (~ /~l(i5) i-~--~/ ,~(p) ~ ~ p r v * ,. / r V * _ ,.' , r-1
r+1 F(p) l 1 F(p) , 1 11 0
~!:::. v vfJ~ / ,~vf1~ /',' ~ ~(p') ,' FCi'') /
111 J~ / ./ vj t~ /
f,;
(p") FCi'")
v1t~ vir 00 00
0 tv:>
§V A Molecular Formula and a Structural Formula of Complexes,
On the aboye chapter, we have expressed a molecular
weight. But now, we shall unify a weight of all comple
xes. In the theory (§I,II,III) we can collect the next
theorem:
(Theorem X) "In the space E; n-polygon has a weight
(n-2) _, (n'-"'3)
* m-rolling polygon has a weight (n-l).
In the space E] all complexes have a
weight:
F=2A.-4m All manifolds have a molecular weight:
* * F=2A.-4x(M).
In the space En~ all complex have a weight:
F=A.-2x(M)
All manifolds have a molecular weight:
* * F=A.-2x(M).
Thus, we can see that all complexes result from a mole
cular complex of all manifolds by streching groups ~~
Now, we can unify a structure formula and a molecular
formula of all complexes.
-31-
complex a molecular formla
~ J 4-polyhedrom • s4 a sphene =4
[4] 6-polyhedrom S4U2
8-pol,Y.hedrom
12-polyhedrom
20-polyhedrom
(. t S, B: body) no e; H : horn
=12
* F
-32-
a structure. formla·
! * <!) F
6-polygon
7-polygon
T-T 2-trus
T-T-T ~-trus
( T( (T)) *T1a<T1a) (T have a child T) =36
( in the belly . (note: * a molecular weight)
of manifold
1 2 3 i T(T(T( ••• (T) .•• )))
1 2 3 i Tl8(Tl8(Tl8( •.. (Tl8) •.• )))
= 181 (a weight)
-33-
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[2] [7] [3] [3]
T16H2Hj T14 =178 (weight)
-34-
241.
Bibliography
[l] G. M. Smmons;
Introduction to topology and modern a
nalysis. New York (1967) pp 279-334.
[2] S. Mac Lane;
Homology. New York (1967) pp 147-250
[3] Veblen, 0. and Whitehead;
The fondation of differential Geometry.
Cambnidge Uni. Press. (1932) pp 50
[4] G. W. Whitehead;
Generalized homology theories, Trans.
Amer. Math. Soc. (1962) pp 227-250
[5] Milnor;
The theory of Morse. Annals of mathe-
matics studies. Priceton Uni.
(1963) pp 10-100
[6] N. Steenrod;
press
The topology of fiber bundles. Priceton
Uni. Press (1951) pp 210-320
[7] L. S. Pontrijagin;
Foundation of combinatorial topology.
Rochester (1952) pp 50-14
[8] P. 7v~-!)-:..tl-"o7;
t'l:#l~fiiJ~ I. II. III. (1954) II pp 104-142
[ 9] J-~ • 7-].,. ~?}~~f*o ( 197 4) pp 83-106
-35-
[工0]ポン卜 Jレヤーギン
連続群論上下。
[11] ヒルツエプJレフ
代数幾何における位相的方法 (1970)
(工972，)
[12] アヒエゼル グラスマン
ヒJレベルト空間論上下。 (1973)
po qo
pp 147-250
pp 12工-208
p ~ 2 - 5