On the Euler characteristic for piecewise linear manifolds

Physics Letters B 273 ( I99 North-Holland

I ) 95-99

On the Euler characteristic M. RoEek

PHYSICS

LETTERS

B

for piecewise linear manifolds

‘Z

and Ruth

M. Williams

Received

3

I October 1991

The Dehn-Sommerville relations and the corresponding equations for the angle sums are used to derive two expressions for the Euler characteristic of a simplicial manifold, firstly in terms of the numbers of even dimensional subsimplices, and secondly in terms of even-dimensional deficit angles. In each case the coefficients involved are related to the Bernoulli numbers.

1. Introduction

where the second sum is over all even dimensional subsimplices such that

Consider a simplicial manifold K of dimension 2~. It is well known that its Euler characteristic x(“‘) is given by

0’” c a’!? c . c a”/

X(lH)= 1 ,=,I

(_

,

)“v,c’“’ .

(1.1)

where ,V/“‘) is the number of i-dimensional cells in K. It is also possible to express x”“’ in terms of angles in the simplicial complex. Let (j, X-) denote the interior angle at the;-dimensional face a’ in the h--dimensional simplex cr’, normalized so that the unit sphere in all dimensions has volume 1 (for example, the interior angle ( I, 3) at an edge in a regular tetrahedron would be given by cos-’ ( f ) divided by the normalization factor 2~). It can then be shown that il(2,r’ = ~(l+~(-l)‘(o,2i,)(?i,.2i,)...(2i,-,.2i,)). (1.2) ’ Work supported ’ ’

in part by NSF grant No. PHY 89-08495. On leave of absence from Institute for Theoretical Physics. SUNY. Stony Brook. NY 11794-3840. USA, On leave of absence from Girton College. Cambridge CB3 OJG, UK and DAMTP. Cambridge CB3 9EW. UK

0370-2693/91/$

03.50 0 1991 Elsevier Science Publishers

(see refs. [ l-3 ] and references therein ). For a given complex, the numbers N,(‘“) are not all independent; they satisfy a set of linear equations called the Dehn-Sommerville equations [ 4,5 1. These encode the requirement that for a simplicial manifold, the link of any subsimplex is homeomorphic to a sphere in the appropriate dimension. The equations can be used to eliminate the N j7”) for odd i from the expression for x(““, x(211)= i a,+,NJ;“‘, ,=,I

a, = ; (2”_

1 )&,

,

(1.3)

where (B2/j are the even Bernoulli numbers. This is the first main result in this paper. The second is an expression for x”“) in terms of even dimensional deficit angles (see (2.4) below), which follows from the fact that certain sums of interior angles satisfy equations exactly equivalent to the Dehn-Sommerville ones:

B.V. All rights reserved

(1.4)

95

Volume 273. number 1.2

PHYSICS LETTERS B

2. The Dehn-Sommerville equations, angle sums and deficit angles

12 December 199l

mensions is given in terms o f interior angles by Z{4)= ~ (

1-

The D e h n - S o m m e r v i l l e equations [5] may be written as + N~,2'')

,., ( ,=,

1

_

Lo+

N} 2'')

for p=O, 1, 2 ..... 2n. ( F o r p = - 1, the equation reduces to ( 1.1 ) provided N(2[ ') is interpreted a s z { > ' ) / 2. ) Only n of these equations are linearly independent, which means that there is precisely the right a m o u n t of information to solve for the o d d - d i m e n sional N~ >') in terms o f the even ones. In the next section, we show how this leads to formula (1.3) for Z (e''~ in terms o f the e v e n N } 2'') . The angle sum _5;~'~ , is defined as the sum o f all interior angles at p-dimensional subsimplices in a simplex o f d i m e n s i o n q, -(,,7_ p

y

(p, q)

y~ {74 ~

(2.1)

(0,2)-

2

G4 ~ a0

(0,4)

(0,2)(2,4)).

¢72 ~

(2.5)

¢70

We simplify this by using the simplest case of the angle-sum relations, that the angles of a triangle add to ( d i v i d e d by the normalization factor o f 27r). This gives, after a p p r o p r i a t e changes of the orders of summation, 7[

y~

(0,2)=

0-2 ~ ao

Z

~

0-2 0-o c o 2

= ~27r

(2.6)

and

~

Y~ (0, 2)(2, 4)

0-o 0-2 ~ a o r74 =, o-2

(2.2)

a l ; ~ 0-,t

a2 ~4 7, 0-2

with ~a q~('~) q - 1 The angle sum S~,u) is related to X], 2'') in ref. [ 5 ] by a p - i n d e p e n d e n t normalization factor (2p~+ 1 )/2

r (p > ' ) = k 2 " ( 2 n + l )

E

t72 ~ O-0

F ( ( 2 n + 3 ) / 2 ) - , Vc2,,) ,

(2.3)

where k is the space-constant. Now the X~,-'''~ satisfy exactly the same linear relations as the ~ (2,,> [ 5 ] and thus the angle sums _fi,(2n) , must satisfy the D e h n Sommerville equations (2.1), with the convention this time that S~2[') = 0 .

~

=½~

aO

r72

(2,4).

(2.7)

o-2 0-4 ~ o-2

The expression (2.5) f o r z (4) may then be written as

Z(4>=

~

( 1-

--½~(1-2

=

~0-4 2>(70 ( 0 , 4 ) )

~

¢74 ~ 0-2

(2,4))

E {7({)4)--½ 2 1~!4) ' o-o a2

(2.8)

This means that we are able to obtain a relation bei ' ( 2 . ) for even p, analogous to tween the angle sums f~,, the expression f o r z (2''> in terms of '-.v pc2,,) for even p. C o m b i n i n g the new formula for Z (2'') with the angle sum relation leads to an expression for Z (z'') in terms of higher dimensional deficit angles. The (norm a l i z e d ) k-dimensional deficit angle at a simplex cr' is defined by

where ~ ) 4 ) is the four-dimensional deficit angle at a vertex (the "solid deficit" in volume there) and E~4~ is the deficit angle at a triangle (the normalized version of the usual deficit angle in four-dimensional Regge calculus [6] ). O f course the G a u s s - B o n n e t theorem in two dimensions is the classic example of the Euler characteristic written in terms o f deficit angles

e~;"=l-

Z~:)= ~2~/)2) '

3~ ok

(j,k).

(2.4)

c;J

Let us illustrate how this works by looking at the case of four dimensions. From eq. ( 1.2 ), the Euler characteristic in four di96

o-o

where cO ,c2) is the deficit angle at a vertex.

(2.9)

Volume 273, number 1,2

PHYSICS LETTERSB

3. Statement and proof of the main results

2~r(2,1 ~' 21 t = L ( 2 i + 1 ) ,=1\ 21

We obtain the following relationships between the Euler characteristic, the numbers of subsimplices, the angle sums and the deficit angles for a simplicial manifold in 2n dimensions: (i)

tl- i

N~>, ~ =½ 21

(

2(i+j)+

E a/+l ;=o

I

l'~vl2,,~

if,

2i

2{1+1}

"

(3.1) (ii)

12 December1991

Z{>')= L "'+lN~2")'

3.2)

-~

I

~

I=l+

1

(iii)

0=

";+

,

3.3)

a,+, ~ {;2"',

3.4)

IO2,

~2 a

;+'\

i=l+j.

N (2") 2(,+;) •

2i

In the second term, use

(2,0:(:)(:::) and then put i = l + 1 - k + l , j = k -

1. This leads to

"£' (2(I+j)+1)

2N{2"'2;' = {~(2n}

2,

;=0

In the first term, put the relation

t=0

L

21

l~r (2n)

/=(}

;=(1 n

(iv)

Z'2"}=

I

Y~ i= 0

a 2~

(2,,)

2 j - 1 a;,

i>t,

(3.5)

with a~ = 1. The a, are universal (i.e., independent of n ) and are related to the Bernoulli numbers B, by (vi)

2 ai=-(22'-l)B2,. /

2;

) f , 2{,+;)

j\2(/-k+l

Replacing / by j in the second term, and separating the j = 0 part in the first sum, we obtain

where the coefficients a, are defined by

(v) a,=l-~,=

;=, ;,=, -

(3.6)

We shall now prove these relations using the DehnSommerville equations (2.1) and the corresponding equations for the angle sums.

2N{2~"},=(2;4 j-2, [2j+I~]{2(;+J)+I)N~,'~i ,.

+ "~/[

1-½ kZ= l aalv2/._l]J~

l=

2;

The definition of a; in relation (v) makes it possible to simplify this to 2..{.,,} ( 2 I + 1)N~,,, ev ,A_~ = \ 21 i f ' 2{;+,> + "-; ~= a;+, ( 2(;+j)+l~v~2"~2;

Proqfqfrelation (i). Suppose that relation (i) holds for i> 1. It certainly holds for i=n, because then it reduces to

"' (2(I+/;)+1) = Y. a;+l N2{;+h, 1= (}

N~,,} = t ( 2 n + l ) N ~ , ''},

{3.7)

which follows immediately from the Dchn-Sommerville relation (2.1) with p = 2 n - l . We shall now prove that relation (i) holds for i=;. From the Dehn-Sommerville equations with

p= 2I- 1,

as required, and relation (i) has been proved by induction on i.

Proofqf relation (ii). From the definition of Z {2,, (the Dehn-Sommerville relation with p = - 1 ) Z{2,,,_ L N ~ ,2, ; , --

2N~'_",= L t=l

(2i+i) \ 2; N~'''-

L \2;J (2i~N{2"' 2i t.

/=/+ I

In the second term, we use relation (i), assumed true for i>I, to give

i--O

--

L 2--v{2 t2,,,I i--I

L N~"'-½ L ....

(2(i+j)+l]v<-"'

,=o

\

/= l ;=o

2i

where we have used relation (i) in the second term. 97

Volume 273, number 1.2

PHYSICS LETTERS B

We can now put i = k - l + 1, j = l - 1 term, to give

z~'-"' =

x~"'-k

in the second

]~=1 ;=1

M(2n) o2n -- 1 * 2n E £'(2n)__

cs2n

a,

1=0

12 December 1991

which gives

xC" 2 ( k - - l + 1)

-

'

Relabelling k and l by i and j, and separating out the i = 0 from the first sum, we may write this as

E a;+, aE2;~ " ' =

/= 0

;= 0

a,+,x~?'.

The result then follows from relation (i). ;=1

l=1

aj~gj--

Proof of rela{ion (t,i). We show that ( 2 / i ) ( 2 2 ; - 1 ) X Bz, satisfies the same defining recurrence relation (v) as a, and can therefore be identified with it. We use some relations for the Bernoulli numbers taken from ref. [ 7 ], together with the fact that Nielsen's B,, is ( - 1 )"+ IB2,, in more usual notation. After this substitution formula (4), on page 172, gives

J'~T(I'I' "

From relalion (v), this becomes

= ~ a,+,N~ ''~ . /=0

2 ( 2 2 " - 1 )B2" + "-' ,=Y~,\(2n~(22,,-2;_2)B2,,_2~=l, 2sJ

as required, where we have used al = 1.

(3.9)

Proof o[" relation (iii). This follows immediately from the proof of relation (ii) when we recall that the S} 2''~ satisfy the same equations as the N} 2''~, with the understanding that ~-I*,;) = ½ZI2,) , ~v'_-L

SI21 ' ) = 0 ,

, i

a,+, /=0

(

Eel2 '',= E a;+, E

cl 2t

=

2s

B2,,

a~=n-1.

t=O

(;,

1 - y.

~

1 )B2,, + ';~' (22~) ( 2 2 n - 2 s -

E

o-2u • r72t

(2i, 2n)

rr 2n ~72s ~ a 2n

a,+, \ .

2,

E

]= 0

E /=0

a 2'~

We now put i = n - s and use

(2i, 2n)

)

s!?' ai+lS~ ''~

~7 ( 2 ; I ) a,+j~,:i a 2n \ i = 0

Using relation (iii) in the second term, we obtain II- 1

It- 1

Z ai+, aE2 ; e~f"'= i E= 0 ai+,N~"'+ aE2n a,,+lS~,"'. [= 0 Now S~,~''~ = 1 by definition, and so 98

1 )B2,,_2. , = r / .

s=[

)

1-

~21

2(22''-

to obtain 2-(22"-l)B>'=l-½n

=

(3.10)

Add (3.9) and (3.10) to give

,-i

1=,)

,=l

_

Proo.fqfrelation (it,). The definition of the deficit angles (2.4) is used to express the right-hand side of relation (iv) in terms of angle-sums: ai+t

and formula (10), on page 152, is equivalent to

;=, \ 2 i - l J

i (22'-1)B2;'

which is exactly of the required form. The defining relation (3.5) holds for i> 1. For i = 1, a~ = 1, and B, can be shown from the simple definition of the Bernoulli numbers to be -~, so relation (vi) holds also in this special case. The relationship between the a; coefficient and the Bernoulli numbers was suggested by John Conway. In fact, the Bernoulli numbers (although not explicitly identified as such) have appeared before [8] as

Volume 273. number 1,2

PHYSICS LETTERS B

coefficients in linear relations between the various D e h n - S o m m e r v i l l e equations. We conclude this section by stating a n u m b e r of other recurrence relations satisfied by the ai: a,=

~-

f2/-l

,=, k2j_2)a,,

--/=,

2j-1 ,=1 -

a,

'

\ 2 j - 2 J "j"

(3.12)

(3.13)

12 December 1991

tetrahedron, for which there is no explicit formula (see for example ref. [ 9 ] ). Our original motivation for exploring possible new formulae for the Euler characteristic was the hope of finding a combinatoric expression for the Hirzebruch signature in four dimensions by operating on the formula f o r z f4). It seems that the significant part o f z ~4) from the point of view of the signature is the part involving the ~4~ deficit angles, and so a deeper understanding of the (0, 4) angles is essential before further progress can be made.

(3.14)

These are most easily proved from formulae involving the Bernoulli numbers.

4. Conclusion Using the D e h n - S o m m e r v i l l e equations, we have shown how the Bernoulli n u m b e r s enter into expressions for the Euler characteristic in terms of either the n u m b e r of even dimensional subsimplices or the higher dimensional deficit angles. It is a feature of the Bernoulli n u m b e r s that they appear in expressions which are in some sense approximations (for example in finite difference approximations to derivatives). Their appearance here may reflect the idea that a piecewise linear space is in a sense an approximation to a space with a continuously distributed curvature. One word of warning: it is not a simple matter to work out the higher dimensional deficit angles for a discrete space-time where the simplices are not equilateral. For example, evaluation ofG(~4t requires the computation of the (0, 4) angles which are solid angles at the vertices of 4-simplices. Each calculation of a (0, 4) involves finding the volume of a spherical

Acknowledgement We have great pleasure in thanking T o m Banchoff, Bill Thurston, C.T.C. Wall, Mark Galassi, and especially John Conway for discussion and correspondence. We are also happy to thank Gisble Murphy for volunteering to type the manuscript. Ruth M. Williams is grateful for hospitality at the Physics Department, Princeton University, where much of this work was done.

References [ 1] T.F. Banchoff, in: Differential geometry, Proc. Special Year 1981-1982, Universityof Maryland, Progress in Mathematics Series. Vol. 32 (Birkh~iuser,Basel, 1982) p. 34. [2] J. Cheeger, J. Dill'. Geom. 18 (1983) 575. [3] J. Cheeger. W. Mi.illerand R.Schrader, Commun. Math. Phys. 92 (1984) 405. [4] M. Dehn, Math. Ann. 61 (1905) 561. [5] D.M.Y. Sommerville,Proc. R. Soc. Lond, A 115 (1927) 103. [6] T. Regge, Nuovo Cimento 19 (1961) 558. [7] N. Nielsen, Trait6 dldmentaire des nombres de Bernoulli (Gauthier-Villars, Paris, 1923). [8] C.T.C. Wall, Can. J. Math. 18 (1964) 92. [ 9 ] H.W. Richmond, Quart. J. Pure Appl. Math. 31 ( 1902 ) 175.

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