PII: 0040-9383(77)90033-7 Tupology Vol. 16, pp. 99-105. Pergamon Press. 1977. Printed in Great Britain ON LOCAL COMBINATORIAL PONTRJAGIN NUMBERS--I PAUL L. KING (Received 15 November 1974) §I. INTRODUCTION NOVIKOV proved[3] that rational pontrjagin numbers are topological invariants. E. Miller[4] showed that they are the only possible local combinatorial invariants, besides the Euler number, for an oriented triangulated smooth manifold. On the other hand the formula of Baum and Bott[l] suggested that, at least in the complex case, the pontrjagin data could be isolated to arbitrarily small neighborhoods of finite point sets, in a rather natural manner. The purpose of this note is to prove that the (first) rational pontrjagin number of a triangulated 4-manifold is a local combinatorial invariant. (All manifolds are assumed oriented.) This result was proved independently by' the author and by John Morgan (unpublished), using rather similar methods. The method extends readily to show that all pontrjagin numbers are local combinatorial invariants of smooth PL manifolds of any dimension. The proof is in §2. In §3, we apply the result to a few examples. In the remainder of this section, we shall give the idea of our proof by a mildly amusing derivation of the local combinatorial Euler formula for a surface from the Gauss-Bonnet theorem. In this situation there is no need to deal with cubed manifolds, but we do it for analogy with subsequent argument. A manifold with polygonal decomposition is cubed if each k-cell of the decomposition is a P L k-cube. Any triangulated space is cubable by taking as vertices the barycenters of each simplicial face, as edges the segments joining the barycenters of k-simplices with the barycenters of (k + l)-simplices in their stars, etc. Let M be a compact oriented cubed surface with vertices v~, with p~ the number of edges (= number of faces) in the star of v~. Consider the following open cover of M - {o~ }. Corresponding to each (open) face F we take the open neighborhood UF of F bounded in M by the segments connecting the barycenters of the 2-faces of M contiguous to F with the common vertices of such a face and F : i 4 / F / FIG. I . This open cover has the properties that UF n Uv, = 0 unless F and F ' are contiguous, and U v n UF n Uv..= 0 always ( F # F' # F"). Make UF a coordinate patch thus: 99 1 ~ PAUL L. KING 1÷31 //....~\., z / \ i / \ \ I+i ~x~ / ' l INN\ / / \ / I o \ / \ / \ / I-I 2 Fro. 2. T h e n on UF tq UF., ZF = Z,.. + C, and the metric g ' = (\/'L--l/2)dz' ~ is globally defined on M - {v~} (dz' = dzF on UF), making e a c h f a c e F a unit square. T h e metric g ' defines a continuous distance function d' on M. which fails to be s m o o t h at the v,. L e t V~ = { p E M : d'(p, v~)<~}. Coordinatize V~ b y z~ = ( z ' ) "~",, where z' is locally a coordinate on the slit disc V~ - { s o m e edge in st v,} induced b y integrating dz'. T h e n =, e x t e n d s c o n t i n u o u s l y a c r o s s the slit, with v~ = 0 , and defines a h o i o m o r p h i c differential dz. L e t g~ = (~/--~/2)dzT~z,. on V. L e t H; be a radial C ~ bump function on V~ at v; ; that is: Hi is a function of r~ alone (z~ = r, eV(-"°,). Hi -- 1 n e a r v. H~ -- 0 outside V. arH~ -< 0 throughout V. W e then h a v e a metric on M: g =EH~g~ + ( 1 - E H ~ ) g ' . i i N o w the G a u s s - B o n n e t t h e o r e m s a y s t h a t 2 w V ' Z I c ~ ( M ) = fM aa log g (where by a b u s e o f language g = (X/L-~/2)g dz'~z), aa log g being the G a u s s i a n c u r v a t u r e o f (M, g). It is evident that on the c o m p l e m e n t of the V~'s. 0a log g - 0 , so 2 7 r X / - - - l c , ( M ) = ~ f v S d log g. But in V. g = (V~L-'I/2)[H~(r~) + (1 - HdrO)(p,2/16)rP/~-~] dz,-~, which is i n d e p e n d e n t of 0,, so in V, H e n c e _X/-i-l~ a ~ !fl.f_](logg)r, dr, dO,. 0 " 0 1 o g g - 2 L0r~: +r~or~j X/--I [ 02 O_~ ](log g )drMO~ = R : . [r,-~rf+-~ri](logg)dr, where R : ¢ " " , r, ,o gl 2 p i p,/",- 2 : ) 1 , , , = ~ V _ - - ] ' ~ i 2 P, 2" T h e r e f o r e c t ( M ) = ~ 1 - (pal4), which is the local E u l e r f o r m u l a f o r a c u b e d surface. §2. LOCALNESS O F THE PONTRJAGIN CLASS THEOREM. A cubed oriented 4-manifold admits a Riemannian structure determined by the combinatorial structure and whose (first) pontrjagin form vanishes outside arbitrarily small neighborhoods of the vertices. ON LOCAL COMBINATORIAL PONTRJAGIN NUMBERS--I 10l T h e p r o o f mimics the derivation o f E u l e r ' s f o r m u l a in the p r e v i o u s section, e x c e p t that there is no c o m p l e t e l y canonical choice o f metric. T h e idea is simply t h a t if a sufficiently natural metric is c o n s t r u c t e d , the pontrjagin f o r m will vanish nearly e v e r y w h e r e . In local terms, we shall simply arrange that enough Christoffel s y m b o l s vanish. At the o u t s e t we m u s t m a k e certain choices. We fix once f o r all a radial C " b u m p function h ( r ) at 0 on the unit ball in R, thus inducing a b u m p function at 0 on the unit ball in Rn. L e t 6Pk (k = 2, 3) be the (countable) set o f combinatorial i s o m o r p h i s m c l a s s e s o f triangulated k - s p h e r e s . L e t T be a map, also fixed o n c e f o r all, which assigns to e a c h X E ~ a PL.equivalence with the Euclidean s p h e r e S k so as to induce a differential structure on X with r e s p e c t to which its triangulation is smooth. T induces differential and Riemannian structures on the (k + l)-disc which is the c o n e on X; call the induced metric gx. (It will be o b s e r v e d that a p p a r e n t l y T is not a c o n s t r u c t a b l e m a p ; it is f o r this r e a s o n that the p r o o f is not constructive. T exists f o r all k < 7, and f o r k >/7 if 6~ is restricted to s m o o t h combinatorial spheres. T h e r e f o r e the p r o o f o f the t h e o r e m is easily modified to give a result f o r s m o o t h 4 n - m a n i f o l d s , n > 1. T c a n n o t be e x t e n d e d f u r t h e r without the restriction on 5ek, h o w e v e r . ) C o n s t r u c t a metric on M as follows. Mimicking the previous section, c o v e r M ' = M - S k 2 M b y neighborhoods o f the 4-faces no three o f which intersect, and m a p e a c h neighborhood into R ' in such a w a y that each 4-face is m a p p e d to the unit c u b e and that on o v e r l a p s coordinates change b y rigid motions (unitary t r a n s f o r m a t i o n plus translation). L e t g ' be the fiat metric on M ' induced b y these local c o o r d i n a t e maps. As before, g' defines a distance function d ' which e x t e n d s to a continuous distance on M but fails to be s m o o t h on Sk2M. L e t {s,} b e the (open) 2-faces o f M. F o r given s~, c h o o s e local coordinates x ' , y', u', v' on the (open) 4-cubes in st s, so that e x t e n d e d f r o m each c u b e s, = { u ' = v ' = 0; 0 < x ' , y ' < 1}. L e t S , = { O < x ' , y ' < l , u ' 2 + v ' 2 < e } (fixed small ~ > 0 ) , and on S t - s t let x ~ = x ' , y ~ = y ' , u, + X/---Iv, = --+ ( u ' + X/----Iv') "~p' (up to unitary t r a n s f o r m a t i o n ) w h e r e p~ = n u m b e r o f 4-faces in st s, and the _.+ sign d e p e n d s on the orientation o f s,. T h e n t h e s e c o o r d i n a t e s e x t e n d s m o o t h l y a c r o s s sl = {u~ = v, = 0; 0 < x , y~ < 1}. L e t g, be the induced (flat) metric on S , dx~ 2 +. • • + dv~ ~ (note that the a b o v e _ sign b e c o m e s insignificant). L e t {e~} be the edges o f M. F o r given e., let S~ be the triangulated 2-sphere defined b y ]0,1[ x S , = st e~ Iq {p E M : d'(p, e~) < ~'} (fixed small e ' > 2~ > 0). T h e n we h a v e a priori (via the m a p T) a chosen metric gs. on the cone on S~. L e t E~ = {p E st e~: d'(p, e,)<-~'}; then E , has an induced flat p r o d u c t metric g~ = dxo2+(~')2g,.(x~ = x ' coordinate along e,, gs i n d e p e n d e n t of x~). Finally, f o r a n y v e r t e x v let V = {p ~ M: d'(p, v) <~ ~"}, (fixed small ~" > 2~' > 0). T h e n V is b o u n d e d b y a triangulated 3-sphere, w h e n c e V is assigned (via T) a flat metric g~ making it an ~"-ball. Define a partition o f unity on M thus. On each V we h a v e the b u m p function h o ( p ) = h ( d o ( p , v ) / ~ " ) , do the distance on V induced b y go. On E , - { V } we h a v e h . ( p ) = h ( d ~ ( p , e ~ ) / ~ ' ) , and this e x t e n d s o v e r the c o n e f r o m v to E~ O OV b y h a ( q ) = ( l - h ~ ( q ) ) h ~ ( e " q / I q l ~ ) and then o v e r the rest of E~ O V b y zero. On S t - { E , } - { V } define h i ( p ) = h(di(p, s~)/e). On (St 13 E o ) - { V } define h~ on the " p r i s m " f r o m e to a E , 13 S: b y hi(q) = (1 - h,(q))h~(~'q/lql~) and e x t e n d b y zero. T h e n on S~ n V define hi on the cone f r o m v to dV n S, b y h i ( q ) = ( I - h~(q))h,(e"q/lqlo) and e x t e n d by zero. Finally, on C({s,} U {e,} U {v}) define h ' = l - Y . h , - Y . h , ~ - E h ~ on {S~} U {E~} U {V} and h '=- 1 elsewhere. N o w define the metric g on M b y g = h ' g ' + ~ ha~ + ~ hag° + ~ h~go. We claim that on the i a v c o m p l e m e n t o f the neighborhoods V o f the vertices, the pontrjagin f o r m p j vanishes identically. Essentially this is b e c a u s e outside the V the Riemannian structure is flat in s o m e direction. We c o m p u t e explicitly. Evidently on M - { S , } - { E , , } - { V } , g is actually flat. Also o b s e r v e that in local orthogonal coordinates, the only non-vanishing Christoffel s y m b o l s are FI,= (g"/2)c~,,, FIj= (g"[2)O~g,,(i~/), FI~ = - ( g " / 2 ) ~ ( i ~ / ) . T h u s on S t - { E ~ } - { V } , g = dx, ~ + dy, ~ + (h,(r,)+ ( 1 - h,(r,))~6r, P/'-~)(dr,2 + r,:dO, ~) T h e only p o s s i b l y non-vanishing Christoffel s y m b o l s are F~, F~, F,~,. H e n c e the only non-vanishing c u r v a t u r e f o r m R / is R , ' , and this is a multiple o f dr~ ^ d#,. T h e r e f o r e 1 0 2 P A U L L. K I N G P ~ k - ¢ e ~ H v j =- 0. O n E~ - { V}. g = h ' g ' + Y~ h,g, + h,,g. = dx,,'- + G, w h e r e G is i n d e p e n d e n t of x,, H e n c e t h e n o n - v a n i s h i n g Christoffel s y m b o l s a r e at m o s t t h o s e n o t i n v o l v i n g x,. T h u s the p o n t r j a g i n f o r m c a n h a v e no t e r m i n v o l v i n g dx~ a n d so v a n i s h e s . H e n c e p~ v a n i s h e s o u t s i d e the a r b i t r a r i l y s m a l l n e i g h b o r h o o d s V o f the v e r t i c e s t' o f M. T h i s c o n c l u d e s t h e p r o o f . §3. EXAMPLES E x a m p l e 1. A local product vertex v o f a c u b e d 4 - m a n i f o l d is o n e at w h i c h the p o l y h e d r a l s t r u c t u r e h a s t h e f o r m of a p r o d u c t s ' × s" w h e r e s ' a n d s" a r e n e i g h b o r h o o d s o f v e r t i c e s o f c u b e d s u r f a c e s , or e × c w h e r e e (resp., c ) is a n e i g h b o r h o o d of a v e r t e x of a c u b e d c u r v e (resp., 3 - m a n i f o l d ) . In t h e s e c a s e s o n e c a n m a k e s u f f i c i e n t l y n a t u r a l c h o i c e s of m e t r i c s to p r o v e d i r e c t l y t h a t the local p o n t r j a g i n n u m b e r p , ( v ) v a n i s h e s . F i r s t c o n s i d e r the c a s e e × c. L e t e,, be a n y e d g e of c. T h e n in V, s t ea= ,~ = : x e a x v Xa Ya FIG. 3. c o o r d i n a t i z e d as i n d i c a t e d , w h e r e u p to rigid m o t i o n x~ = x ' , y~ = y ' , w~ = ( w ' ) '/"o with q~ t h e n u m b e r of 2 - f a c e s in (st e~ :'1 c). T a k e g~ = dx~ 2 + d y , ~ + (V'-Z-T/2)dw~dw~. O b s e r v e that go is a s m o o t h e x t e n s i o n o f t h e n a t u r a l m e t r i c o n s , = e × e~. In the n e i g h b o r h o o d E of e (e is a c t u a l l y t w o edges at v) we h a v e a m e t r i c ge = dxe2+ dye2+ due"+ dye 2, w h e r e xe = c o o r d i n a t e a l o n g e a n d yu, u~, ve a r e o r t h o n o r m a l c o o r d i n a t e s o f t h e m e t r i c d e t e r m i n e d b y the t r i a n g u l a t e d 2 - s p h e r e 0c, ye = Y u ( y ' , u ' , v'), u e = U u ( y ' , u ' , v'), ve = Vu(y', u', v'). T h i s m e t r i c e x t e n d s s m o o t h l y a c r o s s V, a n d we t a k e it to be go as well. W e t h e n o b s e r v e r e a d i l y that t h e m e t r i c o n V g = h ' g ' + ~ h,g, + ~_~ h~g~ + hegE + h~gv i e ~ ~ e has t h e f o r m g = ( d x ' ) Z + G w h e r e G is i n d e p e n d e n t o f x~. H e n c e p , ( v ) = O. S i m i l a r l y , c o n s i d e r s ' × s". If z ' a n d w ' c o o r d i n a t i z e s ' - v a n d s " - v r e s p e c t i v e l y , with g ' = (V'-L-~/2)(dz'-~7+ d w ' d w ' ) , o n e o b s e r v e s r e a d i l y that f o r n a t u r a l c o o r d i n a t e s o n a 2 - f a c e n e i g h b o r h o o d Si o f s ' x v we h a v e z~ = z ' , w, = ( w ' ) "/p; o n S, of v × s" we h a v e z~ = ( z ' ) "/", wi = w ' ; o n S~ of e ' × e" we h a v e z, = z ' , w, = w ' ( w h e r e e ' a n d e" a r e e d g e s o f s ' a n d s"). In e a c h c a s e g, = (~-L-l/2)(dzZ~ + d w Z ~ ) . M o r e o v e r , e a c h edge is c o n t a i n e d in s ' or s", a n d the m e t r i c s o n the c o r r e s p o n d i n g S~ e x t e n d s m o o t h l y a c r o s s t h e edges. If finally we t a k e zL. = (z')~/L wo = ( w ' ) "~p, g~ t h e b y n o w o b v i o u s m e t r i c , we c o n c l u d e t h a t o n V, g = h ' g ' + ~ h~g, + Y~ h,go + hogo h a s t h e f o r m i a g = Ht(r., p~ )dz~dz~ + H:(r~, p~. ) d w ~ d w , w h e n c e a g a i n e a s i l y p , ( v ) = O. E x a m p l e 2. W e n e x t c o m p u t e t h e p o n t r j a g i n n u m b e r o f the 4-simplex barycenter. T h i s is the u n i q u e v e r t e x t y p e v w h o s e link is t h e b o u n d a r y of a s i n g u l a r 4 - s i m p l e x . T o c o m p u t e p , ( v ) , c o n s t r u c t a c u b i s m of S ~ as f o l l o w s . Begin b y t r i a n g u l a t i n g S ~ as t h e b o u n d a r y of a s i n g u l a r 5 - s i m p l e x . Call the v e r t i c e s o f this t r i a n g u l a t i o n r~°(~ = I . . . . . 6). I n t r o d u c e as a d d i t i o n a l v e r t i c e s the b a r y c e n t e r s v~ k of the k - s i m p l i c e s s~ k of the t r i a n g u l a t i o n (k = I . . . . . 4). An edge of the c u b i s m will be a g r e a t c i r c l e a r c v,)t,~ k - ' (k = 1 . . . . . 4) w h e r e s~ k ' C s,f. A 2-face will be a g r e a t S L r e g i o n v,kr~ k-' v~ k--" t,~ k - ' w h e r e S a k - I s ~ - 2 s , ) (k = 2, 3, 4). C" s:_, C~ 3 - F a c e s a n d 4 - f a c e s are d e t e r m i n e d a n a l o g o u s l y . ( T h e a n a l o g o u s c u b i s m of S 2 is s h o w n in Fig. 3.) O N L O C A L C O M B I N A T O R I A L P O N T R J A G I N N U M B E R S - - - I 1 0 3 \ j FIG. 4. T h e n evidently, e a c h v~ ° and v , ' is a 4-simplex b a r y c e n t e r , all with the s a m e orientation, while each v~k(k = 1,2, 3) is a local product. We thus h a v e o-- p,(S')-- X X X v j + X X t~ ct ~ a = 6 p ~ ( v ) + O + O + O + 6 p , ( v ) H e n c e p j ( v ) = O. E x a m p l e 3. Consider the o r i e n t a t i o n - r e v e r s i n g map $ 3 ~ $ 3 : ( u , v . x , y ) ~ ( u , - v , - x , - y ) ( S 3 = { u 2 + v 2 + x 2 + y 2 = 1} C R ' ) . I f T is a triangulation o f an (oriented) S 3, d e n o t e b y T ' the image of T under this map. Call a v e r t e x type v orientation-invariant if the cubical structure on link v is h o m o t o p i c to (link v)'. T h e d e p e n d e n c e o f local pontrjagin numbers on orientation (c.f. [4]) indicates that f o r an orientation-invariant v, p , ( v ) = 0. We shall verify this directly. G i v e n an orientation-invariant v, c o n s t r u c t a triangulated S 4 b y identifying the h e m i s p h e r e s with copies of a small ball a r o u n d v; f r o m the northern h e m i s p h e r e the e q u a t o r will look like link v as 0 (northern hemisphere) and (link v)' as 0 (southern hemisphere). Identify these via the h o m o t o p y (link v ) = (link v)': ¥ v FXG. 5. C o n v e r t this triangulated S" into a c u b i s m b y the s a m e imposition of b a r y c e n t e r s as in the p r e v i o u s e x a m p l e . T h e v e r t e x t y p e s o f the c u b e d S ' will then be two (oriented) copies o f v, several 4-simplex b a r y c e n t e r s , several local products. But these latter t y p e s of vertex h a v e p, = 0 and p , ( S ' ) = 0, so p , ( v ) = 0 as well. E x a m p l e 4. We c o n s t r u c t a v e r t e x t y p e v with non-vanishing pontrjagin number. We need the following e a s y LEMMA. I r a vertex type v is orientation-invariant, then so is the type o[ the apex vertex o f the barycentric subdivision o f the cone on link v. N o w consider the triangulation of S~-o C~: vertices: (i °, 0, 0) = v , ' a = 0, I, 2, 3 (0, i", 0) = v,," i = (0,0, i " ) = v,, ~ TOP Vol. 16. No. I - - H 104 P A U L L. KING edges: v~Jv~,+;, v j v ~ k j , k = 1 , 2 . 3 , j ~ k a , 13 = O , I , 2 , 3 , a # B 2-faces: t',,Jv~÷,v~ k V~ t Vtj2 b, v3 3-faces: j j k /.'~ L!~+IU o U ~ + I F= i U ~ + I VokV-r m 4-faces: s i k i ,- Ua t~a + I D~fll)~ + I D-¢ etc. This is the a n a l o g of the o c t a h e d r a i t r i a n g u l a t i o n o f S~: FIG. 6. Divide out b y the H o p f map $ 5 ~ p2: ( z ' , z 2, z3) --, [ z ' , z : , z 3] to o b t a i n a c e l l u l a r d e c o m p o s i t i o n of p z : v e r t i c e s : v ' = [ 1 , 0 , 0 ] , e d g e s : v ~ v L +, t)~iVO k 2-faces: v ~ J v ~ + , v , k V~ I DB2 Vy3 3-faces: v.Sv~+,vo~v~+, Va i V ~ + I t) O k V'v m v2 = [0, 1,0], v ~ = [ 0 , 0 , l] c o l l a p s e s to v ~ - - , e d g e with v e r t i c e s v ~, v k (there are f o u r such edges c o n n e c t i n g v j and v ~) ~ 2 - c e l l b o u n d e d b y two e d g e s v Jr k ~ 2 - s i m p l e x with v e r t i c e s v ' , v 2, v ~ c o l l a p s e s to two 2-faces ---, 3-cell b o u n d e d b y f o u r 2-cells, one v~v k, one vSv m, t w o dvkv": ( FIG. 7. 4-faces j , j k ~ " 4-cell b o u n d e d b y f o u r V,, L ~ + t L ' B V B + t t ) v 3-cells of the a b o v e t y p e (one p o t e n t i a l b o u n d i n g 3-cell having c o l l a p s e d to two 2-cells) A 4-face is b o u n d e d by an S 3 w h o s e s t e r e o g r a p h i c p r o j e c t i o n to R 3 l o o k s like this: ON LOCAL COMBINATORIAL PONTRJAGIN NUMBERS--I FIG. 8. 105 It is verified that exactly sixteen such 4-cells cover p2 (without overlaps). One also verifies that the vertices v ~ each have the link type of a product: FIG. 9. which in particular is orientation-invariant. Next convert this decomposition to a triangulation by barycentric subdivision, and the triangulation to a cubism by the method of example 2. By the lemma, the original three vertices have vanishing pontrjagin numbers. The link type of the vertices created by barycentric subdivision does not change under the conversion to a cubism. On the other hand, each of these barycenters except the barycenters of 4-cells has the link type of a product and hence pontrjagin number zero. Finally, just as before, the vertices created in converting the triangulation to a cubism all have pontrjagin number zero. Therefore the only nonvanishing pontrjagin numbers might be associated with the sixteen isomorphic types of the barycenters of the original 4-cells. But pl(P 2) = 3. Therefore, a vertex whose link is the barycentric subdivision of the S 3 of Fig. 8, has pontrjagin number 3/16. Remark that as a result of this computation, the link type in question is not orientation-invariant. The reader may verify this directly quite easily. In particular, notice that the antipodal map on the link has the same effect as rotation through ¢r around the horizontal (in the figure) axis. REFERENCES I. P. BAUM and R. Bo'rr: On the zeroes of meromorphic vector fields, Essaies... dddids d Georges deRham, pp. 29-47. Springer, New York (1970). 2. F. HIRZEBRUCm Topological methods in algebraic geometry. Springer, New York (1966). 3. S. KOBAYASm and K. NoMIzu: Foundations of differential geometry, v.2. Interscience, New York (1963). 4. E. Y. MILLER: Thesis, Harvard University (1973). 5. S. P. Novlgov: Rational pontrjagin classes. Homeomorphism and homotopy type of closed manifolds I, Izr. Akad. Nauk 29 (1%5), 1373-1388. 6. R. TnOM: Les classes caract~ristiques de pontrjagin des vari6t6s triangul6es, Syrup. Int. de Top. Alg. (1956), 54-67. University of North Carolina Chapel Hill