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dil eR hePROCEEDINGS=.
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Volume 46 (2015)

RsPages 33-43
E-Published

BRonApril

4, 2014

SEVERALTHE

REMARKSCOMBINATORIAL

MASAHARUONHODGESTARTANABEABSTRACT. In 2007, Scott O. Wilson defined the combinatorial
Hodge star operator % on cochains of a triangulated manifold.
This operator depends on the choice of a cochain inner product,
but he showed that for a certain inner product it converges tothe smooth Hodge star operator as the mesh of the triangulation

tends to zero. He also stated that %2 # +Id in general and raised
a question if %2 approaches +Id.
In this paper, we solve thisproblem affirmatively. We also give a remark about the definitionof holomorphic l-cochains given by Wilson using %.

1.
For cochains equipped

INTRODUCTION

with an inner product,

Scott O. Wilson defined

the combinatorial Hodge star operator ¥% in [7]. The definition is formallyanalogous to that of the smooth Hodge star operator on differential formsof a Riemannian manifold. By using a certain cochain inner product whichhe named the Whitney inner product, he showed that the combinatorial
Hodge star operator, defined on the simplicial cochains of a triangulated
Riemannian manifold, converges to the smooth Hodge star operator as
the mesh of the triangulation tends to zero. A precise statement for thisis Theorem 2.12 and in more detailed form Theorem 2.13. These theorems

are given by using a map from cochains into differential forms, defined byHassler Whitney [6]. Jozef Dodziuk [2] and Dodziuk and V. K. Patodi [3]stated the approximation properties of this map.
and Theorem
2010

These are Theorem 2.6

2.7.

Mathematics

SubjectClassification.

Primary

58A14.

Key words and phrases. cochains,(©2014 Topology Proceedings.

Hodge

33

star.

57R12;

Secondary

58A10,

36

M. TANABE

Definition 2.5. Let K be a triangulation of a Riemannian manifold M.
The mesh nn = n(K) of a triangulation isn = supr(p,q),
where r means the geodesic distance in M and the supremum
over all the pair of vertices p,q of a 1-silnplex in K.

is taken

Theorem 2.6. Let M be a Riemannian manifold with triangulation Kof mesh 1. There exist a positive constant C and a positive integer m,independent of K, such thatjw = WRw|| <C-|[(Id + A)" wl - 7
for all C*>° differential forms w on M.
For the comparison of the Hodge decomposition of a smooth form w
and the combinatorial Hodge decomposition of Rw, we have the following.

Theorem 2.7. Let w € AJ(M) and Rw € C’(K) have Hodge decompositions
w = dw) + wo + dws,
Rw=da,+a9+d*as.

Thenldwy — Wéay|| < C- [[(Id + A)" w|| lwa = Wael <C-[[(Id + A)"w]| +n,|d*ws — Wé*as|| < C-|(Id + A)" w|| - n.
where C and m are independent of w and K.

For details, see [2], [3], and [7].
Whitney [6] also defined a product operation on C(K).

Definition 2.8. We define U: C7(K) ® C*(K) —» C7**(K) bycUT=R(WaoAWT).
We see easily that 6(cUT) = §oUT+(—1)/0UéT and oUT = (—1)7*7U0.
Wilson defined the combinatorial Hodge star operator in [7] and showed
that, for the Whitney inner product, this operator converges to the smooth
Hodge star operator as the mesh of the triangulation tends to zero. For

the rest of this section, all of the statements are attributed to Wilson (]7],8])Definition

2.9.

Let K be a triangulation of a closed oriented manifold

M of dimension n with simplicial cochains C = p, C’.

Let (, ) be an

SEVERAL REMARKS ON THE COMBINATORIAL HODGE STAR37

inner product on C such that C* is orthogonal to C7 for i # j. For o € Cc’
we define «wo € C™*77 by(ko, 7) = (0c UT)[M],
where [M| denotes the fundamental class of M. We call % the combinatorial Hodge star operator.
Several properties of the combinatorial star operator are given below.
Lemma

2.10.

The following hold:(1) 6 =(—1)*16*%, i.e.. % is a chain map.
(2) Foro € CV and € C7, (ko, 7) = (=1)I("=3)(0, %T), i.e, Kk
is (graded) skew-adjoint.

|(3) % induces isomorphisms HC? (K) — HC" ?(K) on harmoniccochains.
From now on, we work under the assumption that the inner producton C is the Whitney inner product unless otherwise mentioned. Let 7

denote the orthogonal projection of L2A7 onto the image of C?(K) underthe Whitney map W.
Lemma

2.11.

Wx=7+W.

This lemma is the key in showing the following theorem which statesthat % converges to x as the mesh 7 tends to O.
Theorem 2.12. Let M be a Riemannian manifold with triangulation Kof mesh n. There exist a positive constant C and a positive integer m,
independent of K, such that| * w—WxRw||for all C*°<C-||(Id+ A)"w|| -ndifferential forms w on M.

This approximation respects thc Hodge decompositions of A(M)andC(K).
Theorem

2.13.

Let M

be a Riemannian manifold with triangulation Kof mesh 1. Let w € AJ(M)and Rw € C?(K) have Hodge decompositions

w = dwy
Rw+ we + d* ws,= da; + az + 6% as.

There exist a positive constant C and a positive integer m,of w and K, such thatindependent| x dwy — Wikéda,|| < C- (||(Id + A)" w|| + [|(Id + A)" dw, ||) - 7.
| x we — Wikaz|| < C-([|(Id + A)" wl] + ||(Id + A)
ws ]|) - 7,
| x d*ws — Wké™as|| < C- (J|(Id + A)" wl} + ||(Id + A)™d*ws]|) - 0.

38

M. TANABE

We remark that we can take a common integer m in Theorem 2.6,Theorem 2.7, Theorem 2.12, and Theorem 2.13.
Wilson defined holomorphic 1-cochains on Riemann surfaces by using
the combinatorial Hodge star. Holomorphic 1-cochains have several properties analogous to holomorphic 1-forms. For details, see [8].

3. APPROXIMATIONTHEOREMS

Recall that the smooth Hodge star x on j-forms satisfies
x2 = (—1)7(n=9)14,
where 7 is the dimension of the manifold. Thus, by the next theorem, wesee that %? converges to +Id as the mesh 7 of the triangulation tends to0.
Theorem 3.1. Let M be a Riemannian manifold with triangulation Kof mesh n. There exist a positive constant C and a positive integer m,independent of K, such that| ¥* w— W?Rw|| <C-||(Id + A)™w| - n
for all C°°
Proof.

differential forms w on M.
Using the triangle inequality and Lemma2.11, we calculate| ** w — We? Rwl||<<|I1%* w= x*WRw|| + | ** WRw — W%2Ruw||||w—WRw| + | ** WRw — xW% Rw|| + | * Wk Rw — W¥*Ruw|=||w—-WRw| + ||* WRw —Wx Rw| + | * Wk Rw — mx

We Ruwl|.

The first term is bounded bylw —WRw|| <C-|(Id + A)"w]| -n,
using the estimate in Theorem 2.6. For the second term, we have|

*WRw —Wx Rw||<|[[*WRw—*w| + ||*w—=|WRw—w|+ | *w—Wx Ruw|

Wx Ruw|,and these are bounded by

IWRw — w| + | *w —Wk Ruw||< C-l(ld+A)"w]-n+C-||(Id + A)" w]| - 7,

SEVERAL REMARKS ON THE COMBINATORIAL HODGE STAR

39

using the estimate in Theorem 2.6 and the estimate in Theorem 2.12. For

the last term, we estimate| * Wk Rw — mx<<We Rw|||| *WxRw—-WR** uw| *WkRw—+*w|| + | ** w—- WR?

= |WkRw -»w|| + lw ~ WRuw|<w||

C-(Id+A)"w||-n+C- [Id+ A) wll - 7.

0The approximation also respects the Hodge decompositions of A(M)and C(K).
manifold with triangulation K

be a RiemannianTheorem

3.2.

Let Mof mesh.

Letw € AI(M)and Rw € C?(K) have Hodge decompositions

w = dw)=Rw+ wg + d* ws,da;+ag+d*as.

and a positive integer m,

There exist a positive constant Cof w and K, such thatindependent|? dun — Wk2daa|| < C- (||(Id + A)"w|| + [|(Id + A)" dwn |) - n,| #2 wp = WikZas]| < C- (|(Id + A)"w]| + [1d + A) wn) - 7.

| %2 d*ws — W?6*as||

Proof.

< C- (||(Id + A)"w|| + |(Id + A)" d*ws]|) - .

For the first statement, we calculate
I yedwn—IW26a,<N12? dw; — $2Wéay || + | ¥> Wéa, — Wk 26a,<|ldwy — Wéay| + | x Wéay — xWda, ||+|| x Wkda; — Wxk26a4||=|ldw; — Wéa|| + | * Wa,+l* W¥

da,— mx— Wkida,||Wxkdaq|.

The first term is bounded by using the first estimate in Theorem 2.7. For
the second term, we have

| * Wéa, — Wikda,|| < | x Way— xdwi|| + | x dw — Wikéa, |,40

M. TANABE

and these are bounded by using the first estimate in Theorem 2.7 and thefirst estitnate in Theorem 2.13. For the last term, we estimate| «Wada,—mxWkdar| < [|x Wahdar — WR" dw |<|x Waka;=|[Wekda, — *dwi| + ||dwir — WR dw, |].

— «> dw || + | ¥* dw — WRdw,||These two terms are bounded by using the first estimate in Theorem 2.13
and the estimate in Theorem 2.6.
The same computations as above lead us to the last two inequalities
by using the latter two inequalities in Theorem 2.7 and Theorem 2.13 andTheorem 2.6 applied to we and d*ws3, respectively.
[1Lieven Smits [5] showed that, on surfaces, ||W§*Rw — d*w|| converges
to 0 for all C* differential 1-form w under a certain restriction on thetriangulations. Recently, Smits’s result was extended to arbitrary dimensions, showing that the above convergence is valid for arbitrary dimension
under a certain mesh condition and showing that this mesh condition isnecessary [1].
Wilson [7] observed that ||WJRw — dw|| converges to 0, and he also
observed that, in short,+5"%=1) —+d"x =xd.

He also raised a question if either of d% or %4* provide a good approximation to dx or xd*, respectively. This question seems to be still open.

4. ON

THE DEFINITIONOF HOLOMORPHIC

COCHAINSIn this section, we fix M to be a topological surface. Wilson |8| definedholomorphic cochains for surfaces using the combinatorial star operator.
Then he introduced combinatorial period matrices which are the period

matrices of holomorphic cochains and gave some (Riemann) bi-linear relations that the periods satisfy, and he proved that for a triangulated
Riemannian 2-manifold (or a Riemann surface) and a particularly nicechoice of inner product, the combinatorial period matrix converges to the(conformal) Riemann period matrix as the mesh of the triangulation tends
to zero.

To define holomorphic

1-cochains, we need to extend some of our defi-nitions to the case of complex valued cochains. Let (, ) be any hermitian
inner product on the complex valued simplicial 1-cochains of a triangulation K for a topological surface AM. We define the associated combinatorial star operator % by(ko, 7) = (ec UT)[M],SEVERAL REMARKS ON THE COMBINATORIAL HODGE STAR

41

where the bar denotes complex conjugation and U is as in §2, extended
over C linearly. Just as with real coefficients, the Hodge decomposition
with complex coefficients holdsCHK) =46C(K)®d H' (K) ® §*C*(K),where H! is the space of complex valued harmonic 1-cochains.
By Lemma 2.10, % induces an isomorphism of H' and is skew-adjoint.
Since % induces an isomorphism of H!, this map admits a unique polar
decomposition w = HU where H is positive definite hermitian and U 1s
unitary. Since % is skew-adjoint, so is U, and thercfore the eigenvaluesof U are +i.
Wilson defined holomorphic

1-cochains as follows.

Definition 4.1 (8, Definition 6.1]). Let X and (, ) be as above.

Let% denote the map on complex valued harmonic cochains, as in Lemma
2.10. with polar decomposition % = HU. The subspace of holomorphic

1-cochains H'°(K) is defined to beH'Y(K) = {we HY(K)|Uw = —iw}.
The subspace of anti-holomorphic 1-cochains H°!(K) is defined to be

HOY K) = {weHY (K)|Uw = iw}.

Subsequently, he remarked the following.

Remark 4.2 (|8, Remark 6.2]). An equivalent definition is to let H1?(K)be the spanof the eigenvectors

for non-positive

imaginaryeigenvaluesof % and let H%!(K) be the span of the eigenvectors for non-negative
imaginary eigenvalues of ¥.
Then he stated the following lemma.

Lemma 4.3 ([8, Lemma 6.3]). Let K be a triangulation of a surface Mof genus g. A hermitian inner product on the simplictal 1-cochains of K
gives an orthogonal direct sum decomposition

HY(K)=H""(K)® HH" (K).
Eachsummandon the right has complex dimension g and complex con-jugation maps H'C(K)to HY(K) and vice versa.

As Wilson indicated, the decomposition follows from the property ofskew-adjoint

operators;

thatof distinct eigenvalues

areorthogonal.

However,complex conjugation does not map H'%(K)is, eigenspaces

toHO (K) in general.
For example, let K be a triangulation of a torus M and let q and b beR-valued harmonic 1-cochains which satisfy(aU b)[M] = 1.

42

M. TANABE

Then a and b form a basis forH'(K). We put
io1=V2a+—b,1o,=—=b.

Let {, ) be a hermitian inner product for which oy, 02 is an orthonormal
basis ofH'(K). Then the matrix representation of % with respect to thebasis 01,02 182t
1—10

the eigenvalues are (1++v/2)i and (1—+/2)i, and the corresponding eigen5\5vectors are constant multiples of ( (1+ va) ) and ( (1 va) ) respectively.

Remark 4.2 definesHK)= fe(1 ~ Va) ) CE c|HONK)= {e((a + V2)and) CE cf.

We see that complex conjugation does not map H1'9(K) to HY1(K).
The main result of [8] is that the combinatorial period matrix converges
to the (conformal) Riemann period matrix as the mesh of the triangulationtends to zero. To show this, Wilson used Riemann’s bi-linear relations

below (see |8, Theorem 6.5]). Let {a;, :-,a,4,b1,---,by} be a canonicalhomology basis for M. (Here “canonical” means that the intersection of
any two basis elements is non-zero only for a; and b;, in which case it

equals one.)
Definition 4.4. For h € H'(K), the A-periods and B-periods of h are
the following complex numbers:

Aj =h(a;)

andB;=h(b;)

forl1<j<g.

Theorem 4.5 (Riemann’s bi-linear relations). If o, 0’ € H°(K)have

A-periods Aj, A’ and B-periods Bj, B;, respectively, theng>(A;B} — B;A}) =0.
j=1
In the proof of Riemann’s bi-lincar relations, Wilson used the statementthat complex conjugation maps H!:%(K) to H*!(K).

Thus, to hold Wil-son’s results, the operator ¥ must be R-valued on R-cochains.

Then, if 7

is a C-cochain, it is straightforward to check that ¥7 = iA7 (A € R) implies 7= —iAT, which means that complex conjugation maps HIO(K)to HY (K).

SEVERALREMARKSONTHECOMBINATORIAL

HODGE

STAR43

By the definition of the combinatorial star operator(ko, 7) = (c UT)[M];% is R-valued on R-cochains if and only if the hermitian inner product isR-valued on R-cochains. Thus, we need one additional assumption, thata hermitian inner product on the cochains to be R-valued on R-cochains.
This assumption is natural and, of course, the Whitney inner productsatisfies it.
Consequently, we offer the following definition.

Definition 4.6. Let (, ) be a hermitian inner product on the complex
valued simplicial 1-cochains which is R-valued on R-cochains. We define
HIO(K) to be the span of the eigenvectors for non-positive imaginaryeigenvalues of % and H%!(K) to be the span of the eigenvectors for nonnegative imaginary eigenvalues of ¥.

Then complex conjugation maps H!1'%(K) to H!(K),and all of thestatements of Wilson's paper [8] hold.
REFERENCES[1]

Douglas Arnold, Richard Falk, Johnny Guzman, and Gantumur Tsogtgerel, On
the consistency of the combinatorial codifferential. To appear in Transactions ofthe American Mathematical Society. Available at arXiv:1212.4472v2 [math.DG].
[2]

Jozef Dodziuk,

Finite-difference approach to the Hodge theory of harmonic forms,

Amer. J. Math. 98 (1976), no. 1, 79-104.

[3] J. Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1-52 (1977).
|4]

BenoEckmann,plex, Comment.
[5]

LievenSmits,Harmonische

funktionen

undrandwertaufgaben

in einemkom-Math. Helv. 17, (1945), 240-255.
Combinatorialapproximation

to thedivergenceof one-forms

onsurfaces, Israel J. Math. 75 (1991), no. 2-3, 257-271.
[6] Hassler Whitney, Geometric Integration Theory. Princeton, N. J.: Princeton University Press,
[7]

1957.

Scott O. Wilson,Cochain algebra on manifolds and convergence under refinement,

Topology Appl. 154 (2007), no. 9, 1898--1920.
[8] ——, Conformal cochains, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5247—
5264.
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