THE TRIANGULATION OF LOCALLY TRIANGULABLE SPACES1
BY
JAMES MUNKRES in Ann Arbor, Michigan
We shall be concerned with the following triangulation problem: "May a space which can be triangulated locally be triangulated in the large?" A locally triangulable space is a separable metric space every point of which has a neighborhood which is homeomorphic to an open subset of some finite polyhedron. Special cases of this problem include the triangulation problems for manifolds, for manifolds with boundary, and for differentiable manifolds; these problems have been attacked in the past, with varying degrees of success.
I n 1935, G. NSbeling published an argument for the triangulation theorem for manifolds [7], but it contained an essential error [9]. Subsequently, S, S. Cairns proved triangulability for differentiable manifolds [3], [4], and T. Rad6 triangulated the general manifold of dimension two [8]. More recently, E. E. Moise proved t h a t three-dimensional manifolds are triangulable [5], and both Moise and R. H. Bing extended this to three-dimensional manifolds with b o u n d a r y [6], [2]. I n the present paper, the author uses some of these results to prove triangulability for locally triangulable spaces of dimension three or less.
We attack the general triangulation problem b y attempting to reduce it to the trian- gulation problem for n-manifolds with boundary. This approach proves successful for n not greater t h a n three. The proofs, while complicated, are elementary in the sense t h a t they involve no algebraic topology.
I n Chapter I certain basic definitions and lemmas are given. A new definition of locally polyhedral space is introduced in Chapter I I , and some of the implications of this definition are studied. I n Chapter I I I a space X*, called the composition space of X, is defined; it is in some respects the opposite of a decomposition space. The technical device of passing from a space to its composition space enables us, in Chapter IV, to treat the general tri- angulation problem.
1 This paper is part of a thesis submitted to the Graduate School of the University of Michigan in partial fulfillment of the requirements for the degree Doctor of Philosophy. The thesis was written under the direction of Prof. E.'E. Molse; the author was a National Science Foundation fellow at the time. Presented to the American Mathematical Society November 12, 1955. The author wishes to thank the referee for his comments.
6 8 JAMES MUNIKRE S
CHAPTER I I n t r o d u c t i o n
t . T h r o u g h o u t t h i s p a p e r , e v e r y s p a c e is H a u s d o r f f a n d e v e r y m a p is c o n t i n u o u s , unless s p e c i f i c a l l y s t a t e d otherwise. T h e closure of a s u b s e t A of X is d e n o t e d b y e i t h e r
~i or C I ( A ) . T h e frontier of A is t h e set CI(A) ~ C l ( X - A ) ; i t is d e n o t e d b y F r ( A ) . T h e w o r d simplex m e a n s " o p e n s i m p l e x " , a n d complex m e a n s " l o c a l l y f i n i t e s i m p l i e i a l c o m p l e x " . The boundaryof a s i m p l e x s is t h e set g s; i t is d e n o t e d b y Bd(s). T h e polytope I K / o f a c o m p l e x K is t h e s p a c e w h i c h is t h e u n i o n of t h e s i m p l i e e s of K ; where no c o n f u s i o n will r e s u l t we do n o t d i s t i n g u i s h b e t w e e n K a n d I K I . T h e k-skeleton of K is t h e c o m p l e x c o n s i s t i n g of all s i m p l i c e s of K h a v i n g d i m e n s i o n k or less; i t is d e n o t e d b y K (kl. A subdivision of K is a c o m p l e x L such t h a t ILl = [ K 1 a n d ] K (k' I c IL (k)] for e v e r y k. I t is a basic p r o - p o s i t i o n t h a t t w o s u b d i v i s i o n s of t h e s a m e c o m p l e x h a v e a c o m m o n s u b d i v i s i o n . A poly- hedron in K is a set which is t h e p o l y t o p e ILl of a s u b c o m p l e x L of s o m e s u b d i v i s i o n of K . I f A is a s u b s e t of I K I , t h e n t h e star of A in K is t h e u n i o n of t h o s e simplices s of K such t h a t some face of s is c o n t a i n e d in A; i t is d e n o t e d b y St(A).
A l i n e a r m a p of one c o m p l e x K i n t o a n o t h e r L is a m a p / o f [K[ i n t o ILl w h i c h m a p s each s i m p l e x s of K i n t o a s i m p l e x t of L in such a f a s h i o n t h a t t h e m a p is l i n e a r w i t h r e s p e c t to t h e b a r y c e n t r i e c o o r d i n a t e s of s a n d t. I f [ is a h o m e o m o r p h i s m of I K I o n t o
ILl a n d b o t h ] a n d [ 1 are l i n e a r m a p s , t h e n ] is a linear isomorphism.
A n n-cell is a h o m e o m o r p h of a n n - s i m p l e x s; a closed n-cell is a h o m e o m o r p h of g.
A n n-mani/old is a s e p a r a b l e m e t r i c space M such t h a t each p o i n t of M has a n e i g h b o r h o o d which is a n n-cell. A n n-mani/old with boundary is a s e p a r a b l e m e t r i c space M s u c h t h a t e a c h p o i n t of M has a n e i g h b o r h o o d whose closure is a closed n-ceil. T h e boundary of a n n - m a n i f o l d w i t h b o u n d a r y consists of t h o s e p o i n t s x of M such t h a t x has no n e i g h b o r h o o d which is a n n-cell; i t is d e n o t e d b y Bd(M). (Do n o t confuse B d ( M ) w i t h Bd(s), w h e r e s is a s i m p l e x . ) T h e set M B d ( M ) is called t h e interior of M a n d is d e n o t e d b y I n t ( M ) . 1.1. L E M M A. Let K be a complex and let L be a subcomplex o~ K. Let L' be a subdivision v / L . This subdivision may be extended to a subdivision o[ K without a[]ecting any simplex o / K outside St lL I"
Pro@ L e t K (m i) be so s u b d i v i d e d t h a t L ' U K (m 1) is a c o m p l e x . L e t s be a n m - s i m p l e x o f K L. I f Bds has been s u b d i v i d e d , we m a y e x t e n d t h i s s u b d i v i s i o n t o s b y m e a n s of r a d i a l lines f r o m t h e b a r y c e n t e r of s; otherwise, we n e e d n o t s u b d i v i d e s. The s u b d i v i s i o n o f K d e f i n e d in t h i s m a n n e r satisfies t h e d e m a n d s of t h e l e m m a .
T H E T R I A N G U L A T I O N O F L O C A L L Y T R I A N G U L A B L E S P A C E S 69 2. T r i a n g u l a t i o n s . A triangulation of a space X is a complex K along with a homeo- m o r p h i s m h carrying I K I onto X. W e sometimes refer to "simplices of X " ; b y this we m e a n those subsets of X which are carried into simpliees of K b y h -1. A subdivision of the t r i a n g u l a t i o n (K, h) is a t r i a n g u l a t i o n (L, k) of X such t h a t h -1 k is a linear m a p of L o n t o K. If J is a s u b c o m p l e x of K, let A = h([ J}), and let g = h, I J (the restriction of h to 1J I).
T h e n (J, g) is the t r i a n g u l a t i o n of A induced by (K, h).
T w o triangulations (K, h) a n d (L, k) of the space X are said to be equivalent if h -1 k is a linear isomorphism of L onto K. T h e y are said to be compatible if t h e y have a c o m m o n subdivision.
If (K, h) ,%rid (L, k) are triangulations of the subsets A a n d B of X, respectively, such t h a t (K, h) a n d (L, k) induce triangulations of Y c A N B which are compatible, then (K, h) a n d (L, k) are said t o be compatible on Y. I t is clear (using 1.1) t h a t there are sub- divisions K ' a n d L ' of K a n d L, respectively, such t h a t t h e triangulations of Y induced b y (K', h) a n d (L', k) are equivalent.
2.1. D E F I N I T I O N . L e t [ be a m a p p i n g of A onto B. [ is strongly continuous if it satisfies the following condition: If U c B a n d / i (U) is open in A, then U is open in B. A continuous m a p / induces (1) a decomposition space C whose points are the sets [-1 (y), a n d (2) a m a p F of C o n t o B which is continuous a n d 1 - 1; if [ is strongly continuous, F is a h o m e o m o r - phism ([l], p. 65).
L e t / be a linear m a p of the complex L into the complex K. Then / is strongly linear if it maps each simplex onto one of the same dimension.
2.2. LEMMA. Let L be a complex and let [ be a strongly continuous m a p o/ [L' onto X satis/ying the [ollowing conditions:
(l) [ is 1 - 1 on each closed simplex o [ L . (2) For each x in I L l , / - l ( / ( x ) ) i s / i n i t e .
(3) Let s I and s 2 be any two simplices o / L , and let [i /1~. (a) I] the sets f (81) and [(s2) intersect, then / 2 1 / 1 maps Sx linearly onto ~2. (b) 1 / / m a p s the vertices o/s~ and those o / 8 z onto the same set, then [:zt[~ maps 81 linearly onto 82.
Then there is a triangulation (K, h) o] X and a strongly linear m a p p o / L onto K , such
t h a t h p = [.
Proo[. W e define an a b s t r a c t complex J. I f v a n d w are vertices of K, let, v be defined to be equivalent to w if ] (v) = ] (w). If v is a v e r t e x of K, let v' denote its equivalence class.
The set of vertices of J will consist of these equivalence classes.
L e t s v0... v~ be a simplex of L. Since [ is a h o m e o m o r p h i s m on ~, the classes
t ! i r
v0, ..., vn are distinct. We define the set {v0 . . . v,j to be a simplex of J . Because of (2), J
70 JAMES I~UNKI~ES
is a locally finite a b s t r a c t complex. L e t K denote a geometric realization of J ; if v' is a vertex of J , let v* denote the corresponding vertex of K. There is a natural linear m a p p of L onto K, defined b y m a p p i n g the vertex v into v* and extending linearly; p is strongly linear.
I t follows t h a t p is strongly continuous. We observe first t h a t a subset U of [L I is open if for every simplex s of L, U N ~ is open in ~. (Then ~ - U is closed, so t h a t for every vertex v of L, Cl(Stv) - U is closed. Hence U fl Sty is open for each v, so t h a t U is open.) Then strong continuity follows from strong linearity, since each closed simplex of L is the homeomorphic image of a closed simplex of K.
Now p(x) = p(y) if and only i f / ( x ) = / ( y ) : L e t x and y be points of ILl contained in the simplices s 1 and s2, respectively. L e t Pt = P] st. I f / (x) = / (y), it follows from condition (3a) t h a t / ~ 1 / 1 = p~lpl on s 1 (so t h a t p(x) = p(y)); if p(x) = p(y), the same result follows from (3b) (so t h a t /(x) = / ( y ) ) . H e n c e the decomposition space C of ILl induced b y / is the same as t h a t induced b y p. Since / and p are strongly continuous, t h e y induce homeo- morphisms F and P, respectively, of C onto X and [ K I, respectively. F P -1 is the desired homeomorphism h.
2.3. LEMMA. Let L and / be as in 2.2, except that condition (3b) m a y / a i l . Let, L 1 be the /irst barycentric subdivision o/ L; then the pair (LI, /) satis/ies all the hypotheses o/ 2.2.
2.4. PROPOSITION..Let A and B be closed subsets o/ the space X; let X = A U B. Let (K, h) and (L, lc) be triangulations o~ A and B, respectively, which are compatible on A N B.
Then there is a triangulation o~ X which induces triangulations o/ A and B which are sub- divisions o/ the given ones.
Proo/. L e t K' and L' be subdivisions such t h a t the triangulations (K', h) and (L', k) induce equivalent triangulations of A ~ B. L e t J denote the complex which is the disjoint union of K ' and L'; let / be the m a p of [JI onto Z which equals h on I g ] a n d equals k on ILl. Then / is strongly continuous: L e t U be a subset of X such t h a t / - I ( U ) is open in I J [ . T h e n / - I ( U ) • [ g t is open in ] K t, so t h a t U N A is open in A (h is a homeomorphism).
Similarly, U N B is open in B. Then A - U and B - U are closed, so t h a t X - U is closed and U is open.
I t is easy to check t h a t ( J , / ) satisfies conditions (1), (2), a n d (3a) of 2.2. L e t J1 denote the first barycentric subdivision of J . Then there is a triangulation (K, h) of X and a strongly linear m a p p of J1 onto K such h p = / . I t follows from this equation t h a t (K, h) induces triangulations of A and B which are subdivisions of (K, h) a n d (L, k), respectively.
THE TRIANGULATION OF LOCALLY TRIA~TGULABLE SPACES 71
C H A P T E R I I
Locally Polyhedral Spaces
3. L o c a l l y p o l y h e d r a l s t r u c t u r e s . I n this section we define the concept of locally polyhedral space. The reasons for introducing this notion are utilitarian; it is essential to our approach to the general triangulation theorem.
3.1. D ~ F I N I T I O N . L e t U be an open subset of X, let L be a finite complex, and let h be a homeomorphism of V into ILl such t h a t [L] =Cl(h(U)) and ]L[ - h ( V ) is the polytope of a subcomplex of L. Then the triple (U, h, L) is said to be a polyhedral neighborhood on X. We sometimes refer to a "simplex of U", meaning a subset s of U such t h a t h($) is a simplex of L.
3.2. D E F I N I T I O N . L e t A be a covering of X b y polyhedral neighborhoods. If x c X , the index o / x relative to A, denoted b y IA (x), is the m a x i m u m of the following set of integers:
(dim s I(U, h, L) EA, s is a simplex of L, and h(x) Es}.
For example, let X consist of the coordinate planes in E3, and let A be the collection uf
all
polyhedral neighborhoods on X. Then for the origin, IA = 0, for each point of the coordinate axes, IA = 1, and for every other point, IA = 2.3.3. D E F I N I T I O N . L e t B be a covering of X b y polyhedral neighborhoods satisfying the following condition: I f (U, h, L) E B, and x and y belong to the same simplex of U, then IB (x) = I s (y). Given B, let A denote the collection of
all
polyhedral neighborhoods (U, h, L) on X which satisfy the following condition: I f x and y are two points of the simplex s of U, then IB(x) = I~(y) > dim s. Then A is a locally polyhedral structure on X, a n d B is a basis for this structure. Note t h a t B ~ A, and t h a t I 8 = IA.A locally polyhedral space is a separable metric space X, provided with a fixed locally polyhedral structure A. If X is a locally polyhedral space (the structure A, although fixed, is not usually mentioned), a polyhedral neighborhood on X will always m e a n an element of the structure A, and the index I(x) of x E X will m e a n IA(x).
3.4. LEMMA. Let A be a locally polyhedral structure on X ; let (U, h, L ) E A . Let V be open in U, and let J be a subcomplex o/some subdivision of L such that C1 (h (V)) = I J I and
] J [ - h (V) is a subcomplex o / J . Then ( V, h l V, J) e A.
3.5. LEMMA. Let K be a / i n i t e complex; let s be an m-simplex o / K ; let xEs. Let ~(K) denote the maximum diameter o / a simplex of K . Given ~ 9 O, there is a subdivision K' o / K such that x lies in an m-simplex o / K ' and 2(K') < ~.
72 JAMES MUNKRES
Pro@ The first barycentric subdivision K1 of K is defined inductively, extending to K (n) the subdivison of K (~-1) by means of radial lines from the barycenters of the n-sim- plices of K. We modify this subdivision slightly. First subdivide K (m-l) barycentrically.
Let yCs; consider the r a y beginning at y and going through x. This r a y intersects B d s in a point belonging to some simplex of the subdivision of Bds; let the dimension of this simplex be denoted by/5(y, x). Let z denote the barycenter of s; if/~(z, x) = m - l , subdivide s baryeentrieally. Otherwise, choose a point y close to z such t h a t /~(y, x ) = m - l , and use radial lines from y instead of from z to subdivide s.
Subdivide the rest of K barycentrieally; let this subdivision be denoted b y K". Then x lies in an m-simplex of K"; by choosing y close enough to z, we m a y make 2(K") as close to ~(K1) as we wish. Hence repeated applications of this method of subdivision will give us the desired subdivision K ' of K.
3.6. COROLLARY. Let X be a locally polyhedral space. I / x E X and I ( x ) = m , there are arbitrarily small polyhedral neighborhoods o] x (i.e., elements o/the structure A) in which x lies in an m-simplex.
3.7. LEMMA. Let x and y be two points o/the simplex s o] the complex K. There is a homeomorphism of St s onto itsel/ which carries x into y.
Pro@ Let / be the map of Cl(Sts) onto itself defined as follows: (1) / is the identity on Fr(Sts), (2) [ maps x into y, and (3) / is extended to S t s b y means of radial lines from x and y, respectively. Then h = / ] Sts is the required homeomorphism.
3.8. PROPOSITIOn. Let X be a metric space. I] X may be covered by polyhedral neigh- borhoods, and A is the collection o/ all polyhedral neighborhoods on X , then A is a locally polyhedral structure on X .
Pro@ We need only to show t h a t A is a basis for a structure. Given e > 0, let A (e) be the subset of A consisting of polyhedral neighborhoods (U, h, L) such t h a t U has diameter less than e. I t follows from 3.5 t h a t IA = IA(~). Let x and y belong to the simplex s of U, where (U, h, L ) E A . B y 3.7, there are neighborhoods of x and y which are homeo- morphic, the homeomorphism g carrying x into y. Then 1A (~)(X) < IA (y), for some e, and the proposition follows, by symmetry.
3.9. D E F I n I T I O n . If X and A are defined as in 3.8, A is called the trivial locally polyhedral structure on X. There are structures which are not trivial. Let K be a finite complex and let i be the identity map of ]K I onto itself. The covering consisting of the single polyhedral neighborhood ( I K I , i, K) is a basis for a locally polyhedral structure.
T H E T R I A N G U L A T I O N O F L O C A L L Y T R I A N G U L A B [ ~ E ~SPACES 73 If, for example, K consists of an n-simplex a n d its faces (n > 1), this s t r u c t u r e is n o t the trivial one.
3.10. D E F I N I T I O N . A m a p / of one locally p o l y h e d r a l space X into a n o t h e r Y is said to preserve index if for each x E X, I (x) = I (] (x)).
3.1l. D E F I N I T I O N . A subspace Y of X is said to be admissible if for each p o l y h e d r a l n e i g h b o r h o o d (U, h, L) on X, Y contains every simplex of U which it intersects.
3.12. P R O P O S I T I O N . Let X be a locally polyhedral space. Let Y be a closed admissible subspace o / X . Then Y has a naturally induced locally polyhedral structure relative to which the inclusion map o/ Y into X preserves index.
Proo]. L e t A be the s t r u c t u r e on X. We define a basis B for a s t r u c t u r e on Y. L e t (U, h, L) EA. h (U N Y) contains e v e r y simplex of L which it intersects, so t h a t its closure is t h e p o l y t o p e of a s u b c o m p l e x J of L. We show t h a t ] J ] - h ( U N Y) is the p o l y t o p e of a subcomplex of J . L e t s be a simplex of J c o n t a i n e d in [J] h ( U N Y); let s' be a face of s.
Suppose s ' c h ( U N Y). T h e n since h(U) is open in ILl, s ~ h ( U ) . B y definition of J , s is a face of some simplex of L c o n t a i n e d in h ( U n Y), a n d since h ( U N Y) is closedin h(U), s ~ h ( U N Y). This is a contradiction. H e n c e s' c IJf - h ( U N Y).
T h e n (U N Y, h I Y, J) is a polyhedral n e i g h b o r h o o d on Y; let this n e i g h b o r h o o d belong to B. I t is clear t h a t if x E Y, IB(x) - IA (x). If x a n d y belong to the same simplex of U N Y, t h e y belong to t h e same simplex of U, so t h a t In(X) - - I A(y). T h e n IB(x ) = I B(y). Since B o b v i o u s l y covers Y, B is a basis for a locally p o l y h e d r a l s t r u c t u r e on Y. Relative to this structure, the inclusion m a p preserves index.
4. T h e p s e u d o s k e l e t o n . T h e p s e u d o skeletons of a locally p o l y h e d r a l space X are closed subsets of X which break X u p into pieces which are manifolds in m u c h t h e same w a y as t h e skeletons of a complex K break i K[ up into cells.
4.1. D I ~ r I N I T I O N . I f X is a locally p o l y h e d r a l space, t h e pseudo k-skeleton of X is the subset of X consisting of all points x such t h a t I (x) < k. I t will be d e n o t e d b y X ~ (see [1], p. 400); we show in 4.2 t h a t it is a closed subset of X. I n general, X ~ depends on the particular s t r u c t u r e A; only when A is t h e trivial s t r u c t u r e is X k topologically deter- mined. I.e., in the e x a m p l e of 3.9, the Z-skeleton a n d pseudo Z-skeleton of 1K] coincide.
N o t e t h a t X ~ is an admissible subspace of X, b y 3.3, so t h a t b y 3.12, X ~ has a n a t u r a l l y i n d u c e d locally p o l y h e d r a l structure, relative to which t h e inclusion m a p preserves index.
W h e n e v e r we consider X ~ as a locally p o l y h e d r a l space, we will suppose it is p r o v i d e d with this structure.
74 J A M E S M U l f f K R E S
A word a b o u t dimension is in order. T h e dimension of X is well-defined, since X is separable a n d metric. One shows easily t h a t t h e locally p o l y h e d r a l space X has d i m e n s i o n n if a n d only if (1) for e v e r y p o l y h e d r a l n e i g h b o r h o o d (U, h, L) on X, dim L < n, a n d (2) for some such neighborhood, d i m L = n. I t follows t h a t X k has dimension n o t greater t h a n k, 4.2. P R o ] ~ o s i w i o ~ r Let X be a locally polyhedral space. Then the set X "~ is closed, and X "~ - X m-1 is an m-mani/old.
Proo/. L e t x c X - x m ; t h e n I ( x ) = k > m. L e t (U, h, L) be a p o l y h e d r a l n e i g h b o r h o o d such t h a t x lies in a k-simplex s of U. N o w S t s contains no simplices of dimension less t h a n k, so t h a t S t s ~ X - X m. Since S t s is open in X, X m is closed.
L e t x E X ~ - X ~-1. L e t (U, h , L ) be a p o l y h e d r a l n e i g h b o r h o o d such t h a t x lies in t h e m - s i m p l e x t o f U. N o w S t t is open in X, so t h a t ( S t t ) N X z i s open i n X ~. B u t (St t) N X "~ = t, b y 3.3, a n d t is a n m-cell n e i g h b o r h o o d of x in X ~ - X e -1. H e n c e X ~ - X ~-1 is an m-manifold.
4.3. LEMMA. Let X be a locaIly polyhedral space, let B a basis /or the structure on X , and let ~ (x) be a n y / u n c t i o n on X such that :r is constant on every simplex o / e v e r y polyhedral neighborhood o / B . I / I (x) = ], then there is a neighborhood V o / x in X j such that :r is constant
Olt V.
Proo/. L e t (U, h, L) be an element of B such t h a t x lies in the j-simplex s of U. L e t V = s; t h e n :r is c o n s t a n t on V. Since V = S t s (~ X ~, V is open in X j.
4.4. COROLLARY. A s s u m e the hypotheses of 4.3. Let Y c X be a connected set on which I (x) is constant. Then ~ (x) is also constant on Y .
Proo/. L e t I ( Y ) = j. T h e n Y c X j. The subsets of Y on which ~r (x) is c o n s t a n t are open in Y, b y 4.3. Since Y is connected, there is o n l y one such subset.
4.5. D E F I N I T I O N . L e t X be a locally p o l y h e d r a l space. A t r i a n g u l a t i o n (K, h) of X is said to be a proper triangulation if (K, h) induces a t r i a n g u l a t i o n of X k, for each k. This is e q u i v a l e n t to the r e q u i r e m e n t t h a t I (x) should be c o n s t a n t on each simplex of X u n d e r this triangulation. If t h e s t r u c t u r e on X is n o t the trivial one, there m a y exist n o n - p r o p e r triangulations. Consider the e x a m p l e of 3.9. A n y h o m e o m o r p h i s m g of t KI o n t o itself defines a t r i a n g u l a t i o n (K, g) of [ K I; b u t g m u s t m a p each simplex onto one of t h e same dimension if (K, g) is to be proper.
4.6. PROPOSlTIO~r Let X be a locally polyhedral space o/ dimension n; let M = X - X n- 1. T h e n
(1) M is a locally polyhedral space; it has a naturally induced locally polyhedral structure relative to which the inclusion m a p o / . l ~ into X preserves index.
T H E T R I A N G U L A T I O N OF L O C A L L Y T R I A N G U L A B L E S P A C E S 75
(2) 3~ ~-1 = 3~ - M = F r M .
(3) I] X ~-1 is properly triangulated, this triangulation induces a proper triangulation o] M ~-1.
Proo[. L e t ~(x) d e n o t e t h e largest integer m such t h a t x E C l ( X m - Xm-1). T h e n co(x) is c o n s t a n t on e v e r y simplex of e v e r y p o l y h e d r a l n e i g h b o r h o o d on X. M consists precisely of those points of X for which ~(x) = n. Hence M is an admissible subspace of X, a n d (1) follows f r o m 3.12. I t is clear t h a t (2) holds. To prove (3), let X ~-1 be properly triangulated.
B y 4.4 a n d 4.5, each simplex of X ~-1 which intersects F r M m u s t lie in F r M. Since F r M is closed, it is t h e p o l y t o p e of a s u b c o m p l e x of X n-1. This induced t r i a n g u l a t i o n of F r M is proper, since the inclusion m a p of F r M i n t o X ~-1 preserves index.
4.7. REMARK. The hypothesis t h a t t h e t r i a n g u l a t i o n of X ~-1 is proper is needed in t h e preceding proposition. Consider t h e following example: T a k e two copies of E 2 a n d i d e n t i f y t h e m along t h e closed half-line x > 0, y = 0. T h e n t a k e a n o t h e r closed half-line a n d identify its end point with the end p o i n t of the previous half-line. Consider the resulting space X as a locally p o l y h e d r a l space u n d e r t h e trivial structure. X 1 is the h o m e o m o r p h of a s t r a i g h t line, a n d X ~ is a single p o i n t dividing t h e line into two parts, one of which is F r M. Obviously, if a t r i a n g u l a t i o n of X 1 is to induce a t r i a n g u l a t i o n of F r M , the point X ~ m u s t be a v e r t e x in this triangulation. Hence t h e t r i a n g u l a t i o n of X 1 m u s t be proper.
Less trivial is t h e following example: T a k e t w o copies of E 3 a n d identify t h e m on t h e closed half-plane x > 0, z = 0. Take a n o t h e r c o p y of this closed half-plane a n d identify it with t h e previous half-plane along their y-axes. T a k e t h e trivial locally p o l y h e d r a l s t r u c t u r e for this space X. X ~ is a plane, a n d X 1 is a line r u n n i n g across this plane, dividing it into t w o half-planes. F r M is one of these closed half-planes. I n t h e previous example, F r M was always a polyhedron in Y = X n-l, w h e t h e r t h e t r i a n g u l a t i o n of Y was p r o p e r or not.
I n t h e pre~ent example, this is n o t t h e case; there are clearly triangulations of Y relative to which F r M is n o t a polyhedron.
Our a p p r o a c h to t h e t r i a n g u l a t i o n p r o b l e m will be b y i n d u c t i o n on the dimension of t h e space. W e first choose a t r i a n g u l a t i o n of Y a n d seek to e x t e n d this t r i a n g u l a t i o n to a t r i a n g u l a t i o n of X. I f we should h a p p e n to choose a t r i a n g u l a t i o n of Y relative to which F r M is n o t a p o l y h e d r o n , this would n o t be possible. Hence we shall require t h a t t h e t r i a n g u l a t i o n of Y be a p r o p e r one.
5. S i n g u l a r i t y of a p o i n t . T h e preceding proposition reduces our p r o b l e m to t h a t of
" e x t e n d i n g " a proper t r i a n g u l a t i o n of F r M t o M ; t h e a d v a n t a g e is t h a t 3~ is a manifold w i t h b o u n d a r y except possibly at p o i n t s of F r M. L e t us s t u d y these points more closely.
7 6 J A M E S M U N K R E S
5.1. D E F I N I T I O N . L e t X be a l o c a l l y p o l y h e d r a l space; l e t x 6 X . T h e n x is s a i d t o h a v e singularity ]~ with respect to X m and X if k is t h e s m a l l e s t n u m b e r such t h a t x h a s 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 U such t h a t U - X m h a s ]c c o m p o n e n t s . W e o b t a i n i n 5.5 a n e q u i v a l e n t d e f i n i t i o n w h i c h is m o r e u s a b l e . I t will follow t h a t e v e r y p o i n t of X h a s f i n i t e s i n g u l a r i t y .
L e t X h a v e d i m e n s i o n n. U s u a l l y we a r e c o n c e r n e d w i t h t h e s i n g u l a r i t y of a p o i n t w i t h r e s p e c t t o X ~-1 a n d X . I f we s a y " x h a s s i n g u l a r i t y / c " , t h e p h r a s e " w i t h r e s p e c t to X ~-1 a n d X " will be u n d e r s t o o d . F u r t h e r , x will be s a i d to be non-singular if its s i n g u l a r i t y is less t h a n 2; o t h e r w i s e i t is singular.
L e t M = X - X n 1. T h e s i n g u l a r p o i n t s x of X lie in F r M: I f x is in M , t h e n x h a s a c o n n e c t e d n e i g h b o r h o o d w h i c h does n o t i n t e r s e c t X ~ 1, so t h a t its s i n g u l a r i t y is 1. I f x is n o t in M , i t has a n e i g h b o r h o o d w h i c h does n o t i n t e r s e c t M , so t h a t x has s i n g u l a r i t y 0.
5.2. LEMMA. Let K be a complex in E~, let L be a subcomplex o/ K , and let x be a point:
o/ the simplex s of K . Let S be a spherical neighborhood o/ x in i K I , lying in St s. T h e n each component o/ St s - [L I contains exactly one component o/ S - I L l .
Proof. P r o j e c t St s o n t o S b y m e a n s of r a y s f r o m x. T h i s h o m e o m o r p h i s m c a r r i e s e a c h s i m p l e x i n t o itself, so t h a t i t m a p s St s - ILl o n t o S - I L l . T h e l e m m a follows.
5.3. COROLLARY. A s s u m e the hypotheses o/ the preceding lemma. I / U is any neigh- borhood o/ x in [ K [ which lies in St s, then U - ILl has at least as m a n y components as
ts-ILl.
5.4. R E M A R K . L e t X be a l o c a l l y p o l y h e d r a l space; l e t (U, h , L ) be a p o l y h e d r a l n e i g h b o r h o o d on X a n d l e t x b e l o n g t o t h e s i m p l e x s of U. St s is o p e n in U, a n d C1 (h (Sts)) is t h e p o l y t o p e of a s u b c o m p l e x J of L. M o r e o v e r , ]JI - h ( S t s ) is a s u b c o m p l e x of J . B y 3.4, (St s, h [ S t s, J) is a p o l y h e d r a l n e i g h b o r h o o d on X. T h e s t a t e m e n t " L e t (Sts, ]~, J } be a p o l y h e d r a l n e i g h b o r h o o d of x " m e a n s t h a t i t is d e r i v e d in t h i s w a y a n d t h a t x is in s.
5.5. P R O P O S I T I O N . Let X be a locally polyhedral space; let (Sts, ]c, J) be a polyhedral neighborhood o/ x E X . Let S t s - X "~ have k components. T h e n x has singularity ]c with respect to X m and X .
Proof. G i v e n v > 0, let S be a s p h e r i c a l n e i g h b o r h o o d of h (x) h a v i n g d i a m e t e r less t h a n e a n d l y i n g in St(h(s)). T h e n S - h ( X m) has ]c c o m p o n e n t s , b y 5.2, a n d so does h - l ( S ) - X ~.
B y d e f i n i t i o n , t h e s i n g u l a r i t y of x w i t h r e s p e c t t o X m a n d X is n o t g r e a t e r t h a n ]c.
On t h e o t h e r h a n d , if U is a n y n e i g h b o r h o o d of x w h i c h is s m a l l e n o u g h to lie in S t s , t h e n b y 5.3, U - X m m u s t h a v e a t l e a s t ]c c o m p o n e n t s . H e n c e t h e s i n g u l a r i t y of x w i t h r e s p e c t t o X ~ a n d X is n o t less t h a n / c .
T H E T R I A ~ T G U L A T I O ~ O F L O C A L L Y T R I A N G U L A B L E S P A C E S 77 5.6. COROLLARY. Let X be a locally polyhedral space, and let :r (x) equal the singularity of x with respect to X m and X . Then cr is constant on every simplex o / e v e r y polyhedral neigh- borhood on X .
5.7. L],]MMA. Let X be a locally polyhedral space of dimension n such that X = C1 ( X - x n - 1 ) . Let ] <_ n - 1, and suppose X has no singular points o / i n d e x greater than ]. I / x c X has singularity k (with respect to X ~ 1 and X), then it has singularity k with respect to X ~ and X .
Proof. L e t (Sts, h, L) be a p o l y h e d r a l n e i g h b o r h o o d of x (5.4). T h e n S t s - X ~-1 has k c o m p o n e n t s ; we show t h a t S t s - X j also has k components. N o t e t h a t S t s consists entirely of n-simplices a n d their faces, since X = C I ( X - X ~ 1). L e t y E S t s N ( X ~-1 - X J ) . T h e n y lies in some proper face t of an n-simplex of St s. Since y has index greater t h a n ], y is non-singular, so t h a t St t - X ~-1 has e x a c t l y one c o m p o n e n t , which m u s t be contained in some c o m p o n e n t of S t s - X ~-1. Since St t is open in X, y is a limit point of only one com- p o n e n t of S t s - X n-1.
L e t t h e c o m p o n e n t s of S t s - X ~-1 be d e n o t e d b y Ci (i = 1 , . . . , k). L e t Di = C~ N (Sts - X J ) . E a c h set D~ is connected, consisting of Ci plus some of its limit points. As just shown, t h e sets D~ are disjoint a n d their u n i o n is S t s - X j. H e n c e t h e y are t h e c o m p o n e n t s of S t s - X -~, a n d there are e x a c t l y k of t h e m .
5.8. PROPOSITIOZr Let X be a locally polyhedral space of dimension n such that X = C l ( X - X=-I); suppose X has no singular points. I f x E X has index n - 1 or n - 2, it has a neighborhood whose closure is a closed n-cell, while i / x has index n - 1, it has no neighborhood which is an n-cell.
Proof. L e t (Sts, k, L) be a p o l y h e d r a l n e i g h b o r h o o d of x such t h a t d i m s = I ( x ) . L e t t d e n o t e t h e simplex k (s) of L; t h e n L = Cl (St t), a n d L consists entirely of n-simplices a n d their faces.
Suppose x has index n - 1. L contains only one n-simplex, since otherwise S t t - t = S t t - k ( X n-l) would n o t be connected. T h e n [L] is a closed n-simplex a n d k(x) lies on t h e b o u n d a r y of this closed n-simplex.
Suppose x has index n - 2. Consider, for each n-simplex t t of L, the face e t of t t opposite t. L e t ~Y denote t h e collection of these 1-simplices e~ a n d their vertices. T h e n ] N I is a con- n e c t e d 1-manifold with b o u n d a r y : I f ]N[ were n o t connected, t h e n the n-simplices of L could be divided into t w o disjoint sets such t h a t no simplex of t h e first set has a n (n - 1)- dimensional face in c o m m o n w i t h a simplex of t h e second set. T h e n St t - t = S t t - k (X n-e) would n o t be connected, c o n t r a d i c t i n g 5.7. If [N] is n o t a 1-manifold with b o u n d a r y , t h e n t h r e e 1-simplices of N m e e t at a v e r t e x v. I f et a n d ej m e e t at v, t h e n tt a n d tj h a v e an
78 JAMES MUNKRES
(n - 1)-dimensional face r in c o m m o n ; r is t h e face s p a n n e d b y v a n d t. I f t h e t h i r d n- simplex t~ has r as a face, t h e n t h e points of h -1 (r) h a v e index n - 1, so t h a t S t r - r equals S t r - k ( X ~ - l ) . B u t S t r - r is n o t connected, c o n t r a d i c t i n g the fact t h a t X has no singular points.
Then I N] is a h o m e o m o r p h of a closed line segment or a circle. F r o m this, it is readily shown b y combinatorial means t h a t [LI is a closed n-cell. I n t h e f o r m e r case, x lies on its b o u n d a r y ; in t h e latter case, x lies in its interior.
C H A P T E R I I I
The Composition Space
T h e preceding proposition indicates t h a t M would be more like a manifold with b o u n d a r y if one could dispose of the singular points of F r M . T h e a c c o m p l i s h m e n t of this task occupies the present chapter. We f o r m a new space b y "pulling 217 a p a r t " along the singular p a r t s of its frontier. T h e n e w space has fewer singular points t h a n 217 (7.9), b u t it is equivalent to 2~ as far as t r i a n g u l a b i l i t y is concerned (8.5).
The following a s s u m p t i o n holds t h r o u g h o u t t h e chapter: X is a locally p o l y h e d r a l space of dimension n such t h a t (1) X = Cl ( X - X ~ - I ) , a n d (2) X h a s . n o singular points of index greater t h a n the fixed n u m b e r ] (] _< n - 1).
6. D e f i n i n g t h e s p a c e . L e t Ak denote t h e set of all points of X h a v i n g index ] a n d singularity k, k > 1. L e t A = [.JkAk. W e note certain i m p o r t a n t facts:
(1) If (U, h, L) is a p o l y h e d r a l n e i g h b o r h o o d on X, t h e n Ak contains e v e r y simplex of U which it intersects, b y 5.6.
(2) The sets Ak are disjoint open subsets of X s, b y 4.3.
(3) A = [3Ak. I f x is a limit point of A, each p o l y h e d r a l n e i g h b o r h o o d of x intersects only a finite n u m b e r of t h e sets Ak, b y (1). T h e n x is a limit p o i n t of some Ak.
(4) L e t x E A k , a n d let (Sts, h, J ) be a p o l y h e d r a l n e i g h b o r h o o d of x such t h a t dim s = ]. T h e n S t s - X ~-1 has k components, b y 5.5, a n d so does S t s - X s, b y 5.7.
Also, S t s - X s = S t s - s = S t s - A k .
6.1. LEMMA. L e t x E A k. L e t ( S t s , , ki, J i ) be polyhedral neighborhoods o / x (i = 1, 2, 3).
I / C 1 is a c o m p o n e n t o] S t s 1 - X s, there is exactly one c o m p o n e n t C 2 o / S t s 2 - X s s u c h that C 1 N C 2 has x as a l i m i t point. I / C a is the c o m p o n e n t o / S t s a - X s s u c h that C I N C a has x as a l i m i t point, then so does C~ N C a.
Proo/. This follows i m m e d i a t e l y f r o m 5.2.
T H E T R I A I ~ G U L A T I O I ~ OF L O C A L L Y T R I A I ~ G U L A B L E S P A C E S 79 6.2. D E F I I ~ I T I O N . W e define t h e composition space X*. I t s p o i n t s consist of one c o p y of e a c h p o i n t of X - A , a l o n g w i t h k copies of e a c h p o i n t of Ak (/c = 2, 3 . . . . ).
W e define a f u n c t i o n ~o. Choose for e a c h p o i n t x of A , a f i x e d p o l y h e d r a l n e i g h b o r h o o d (Stsx, lcx, Jx) of x. F o r e a c h d i s t i n c t c o m p o n e n t C~x of Stsx - X j, l e t eo (C~, x) d e n o t e a dif- f e r e n t one of t h e copies of x in X* (there a r e / c such c o m p o n e n t s , a n d ]c copies of x). T h e n if (Sts, k, J) is a n y p o l y h e d r a l n e i g h b o r h o o d of x a n d C is a n y c o m p o n e n t of S t s - X j, t h e r e is e x a c t l y one c o m p o n e n t Cix of Stsx - X j such t h a t C N C~ has x as a l i m i t p o i n t . Define r (C, x) = co (C~, x). B y 6.1, ~o h a s t h e following i m p o r t a n t p r o p e r t y : I f (Sts~, lc~, J~) is a p o l y h e d r a l n e i g h b o r h o o d of x E A a n d Ct is a c o m p o n e n t of S t s ~ - X j (i = 1, 2), t h e n o~ (C1, x) = o) (C2, x) if a n d o n l y if C 1 ~ C 2 h a s x as a l i m i t p o i n t .
L e t H be t h e n a t u r a l p r o j e c t i o n of X* o n t o X . W e define a basis for o p e n sets in X*:
(A) I f U is o p e n in X , ~ - I ( U ) is a basis e l e m e n t . ( H e n c e H is c o n t i n u o u s . )
(B) L e t x E A a n d l e t (Sts, k, J ) be a n y p o l y h e d r a l n e i g h b o r h o o d of x such that d i m s = ] . L e t C be a n y c o m p o n e n t of S t s - s . Since s ~ Ak, w(C, z) is d e f i n e d for e a c h p o i n t z of s. L e t t h e set
W ( C ) = ~ - ~ ( C ) U { ~ ( C , z ) l z ~ s } be a basis e l e m e n t .
These n e i g h b o r h o o d s will be r e f e r r e d t o as n e i g h b o r h o o d s of t y p e (A) a n d of t y p e (B), r e s p e c t i v e l y .
6.3. PI~OPOSITION. X * is a Hausdor// space.
Proo/. W e m u s t s h o w t h a t if y is c o n t a i n e d in t h e i n t e r s e c t i o n of t w o basis e l e m e n t s , so is a basis e l e m e n t c o n t a i n i n g y. S u p p o s e t h e basis e l e m e n t s a r e of t y p e (B), s a y W(C~) (i = 1, 2), w h e r e C~ is a c o m p o n e n t of Sts~ - s~. ( S i m i l a r a r g u m e n t s a p p l y in t h e o t h e r cases.) I f y EH -1 (C1) , t h e n y ET~ -1 (C2) as well. Ci is o p e n in X , since X is l o c a l l y c o n n e c t e d a n d C~
is a c o m p o n e n t of t h e o p e n set Stst - st. T h e n g - 1 (C 1 ~ C~) is t h e r e q u i r e d basis e l e m e n t . Otherwise, y = eo (C1, x) = w (C2, x), for s o m e x in s 1 N s~. L e t (Sts3,/ca, J3) b e a p o l y - h e d r a l n e i g h b o r h o o d of x such t h a t S t s a = S t s 1 N S t s 2 a n d d i m s 3 = ?" ( b y 3.6). L e t C 3 be t h e c o m p o n e n t of S t s a - s a s u c h t h a t co(Ca, x) = y. T h e n W ( C a ) ~ W(C1) N W(C2).
H e n c e X* is a t o p o l o g i c a l space. To s h o w i t is H a u s d o r f f , l e t x, y E X * a n d l e t ~ ( x ) = H (y) = z (the o t h e r case is t r i v i a l ) . L e t (Sts, ]c, J ) b e a p o l y h e d r a l n e i g h b o r h o o d of z such t h a t d i m s = }. T h e n x = co (C1, z) a n d y = co (C~, z), w h e r e C 1 a n d C~ a r e d i s t i n c t c o m p o n e n t s of S t s - s. W(C1) a n d W(C2) a r e d i s j o i n t n e i g h b o r h o o d s of x a n d y, r e s p e c t i v e l y .
6.4. LEMMA. Let y E X * , and let U be a neighborhood o / y . I / x e ( y ) E X - A , U contains a neighborhood o / y o / t y p e (A); i / H ( y ) e A , one o / t y p e (B).
8 0 JAMES MUNKRES
Proof. Let z e ( y ) E X - A. If U contains no neighborhood of y of type (A), there is a neighborhood W(C) of y contained in U. Then yC~-I(C), since ~(y) is not in A, so t h a t
~-1(C) c U is a neighborhood of y of type (A), contrary to hypothesis.
Let ~ (y) EAk. If U contains no neighborhood of y of type (B), there is a neighborhood
~ - I ( V ) of y of type (A) contained in U. There is a polyhedral neighborhood (Sts, k, J) of
~(y) such t h a t dim s = ?" and S t s c V. For some component D of Sts - s , ~o(D, ~(y)) = y.
Then W ( D ) c U is a neighborhood of y of type.(B), contrary to hypothesis.
6.5. PROPOSITION. X* has a countable basis.
Proof. Let U1, U 2 .. . . be a countable basis for X. Then the collection {~-1 (U~)} contains arbitrarily small neighborhoods of every point in ~ I ( X - A ) , b y 6.4. On the other hand, there is a countable collection {(Stsik, g~s, J,s)} of polyhedral neighborhoods on X such t h a t (1) dim sis = ], (2) s i s c A, and (3) the collection contains arbitrarily small neighbor- hoods of each point of A. This collection is defined b y choosing a polyhedral neighborhood (Stsis, gis, Jis) satisfying (1) and (2) such t h a t U , c S t s i s c Us, for each pair of indices i, k for which such a polyhedral neighborhood exists. F r o m each polyhedral neighborhood in this collection m a y be derived a finite number of neighborhoods in X* of type (B); the resulting derived collection will still be countable. Using 6.4, one m a y prove t h a t this derived collection contains arbitrarily small neighborhoods of each point of ~ 1 (A). The union of these two collections of open sets is a countable basis for X*.
6.6. PROPOSITIOn. The map ~ is strongly continuous.
Proof. Let U be a subset of X such t h a t ~ - I ( U ) is open in X*. Let xE U. I f x E X - A,
~-1 (x) has a neighborhood ~-1 (V) of type (A) contained in ~-1 (U). Then V is a neighborhood of x contained in U.
If xEAk, let Yl . . . . , y~ be the points of ~-l(x). There is a neighborhood W(C,) of y, of type (B) contained in ~ - I (U), derived from the polyhedral neighborhood (Sts~, k,, J,) of x.
There is a polyhedral neighborhood (Sts, k, J) of x such that dim s = ] and S t s ~ NSts,.
L e t D, be the component of St s - s such t h a t eo (D,, x ) = y i . Then W ( D , ) ~ W(Ci)
~ - I ( U ) . The set ~-1 (Sts) = U W (D,) is contained in ~ - I ( U ) , so t h a t Sts is a neighborhood of x contained in U.
Hence U is open, and ~ is strongly continuous.
7. I m p o s i n g a l o c a l l y p o l y h e d r a l s t r u c t u r e . We recall some facts about covering maps. If p maps E into B and / maps Y into B, then a m a p / * of the subset Z of Y into E is said to be a lilting of / over Z if for every z EZ, p/* (z) = / (z). A map p of E onto B is said to be a covering map if each point x of B has a neighborhood V such t h a t p maps each of
T H E T R I A ~ G U L A T I O ~ O F L O C A L L Y T R I A N G U L A B L E S P A C E S 81 t h e c o m p o n e n t s of p - l ( V ) h o m e o m o r p h i c a l l y o n t o V. I t is a basic proposition on covering m a p s t h a t if p is a covering m a p a n d / is a m a p of t h e simply connected, arcwise connected, locally arcwise connected space Y into B, t h e n a lifting of f over a single point of Y m a y be u n i q u e l y e x t e n d e d to a lifting of f over all of Y.
7.1. LEMMA. Let f m a p Z into X and let f* be a (not necessarily continuous) map o / Z into X * which is a lifting of f over Z. T h e n / * is continuous at each point o/ f-1 ( X - A).
Proof. This follows i m m e d i a t e l y f r o m 6.4.
7.2. LEMMA. Let W(C) be a neighborhood on X * of type (B), derived from (Sts, h, L).
Then ~ maps W (C) homeomorphically onto C U s.
Proof. Clearly ~ 1 W ( C ) is continuous a n d 1 - ! ; let g d e n o t e t h e inverse map. B y 7.1, g is continuous at each p o i n t of C. L e t x E s. L e t U be a n e i g h b o r h o o d of g (x) = y = co (C, x) in W (C). T h e n U is open in X* a n d contains a n e i g h b o r h o o d W (D) of y of t y p e (B), derived f r o m (St t, k, J). L e t V = D U t; t h e n V ~ C U s, since D ~ C a n d t ~ s. E a c h com- p o n e n t of St t - t is contained in a different c o m p o n e n t of S t s - s, so t h a t V = (C (J s) ~ (St t).
H e n c e V is open in C U s. Since g (V) ~ U, g is continuous at x.
7.3. LEMMA. Let s be a simplex; let h map -~ homeomorphically into X , mapping s into A k. Let s' denote -~ ~ h-l (Ak). Then ~ - l h ( s ' ) has lc components, each o/whose closures is mapped homeomorphically onto h (~) by ze.
Proof. W e show t h a t 7e is a covering m a p of t h e space z - l ( A ) onto A. L e t x E A k , a n d let (St t, ]c, J) be a p o l y h e d r a l n e i g h b o r h o o d of x such t h a t d i m t = j. T h e n t is a neighbor- h o o d of x in A. L e t C i be a c o m p o n e n t of St t - t, a n d let s i d e n o t e W(Ct) N ~ - I ( A ) . T h e n 7~
m a p s si h o m e o m o r p h i c a l l y onto t, b y 7.2, a n d t h e sets st are m e r e l y t h e c o m p o n e n t s of
~-l(t).
L e t z be a fixed point of s', a n d let Yl . . . y~ be t h e points of g - l h ( z ) . Define hi (z) = Yt;
t h e n h~ is a lifting of h over z, so t h a t it m a y be u n i q u e l y e x t e n d e d to a lifting hi of h over all of s'. T h e sets h i (s') are disjoint, for if hi (s') a n d hm (s') h a d t h e p o i n t q in c o m m o n , t h e n b o t h hi a n d hm would be t h e unique e x t e n s i o n t o s' of t h a t lifting of h over h - l ~ (q) which carries this p o i n t into q. B u t t h e y disagree at z. Since t h e sets h i (s') are k in n u m b e r , their u n i o n is ~ - l h ( s ' ) .
N o w hi is defined on s'; if w E - 5 - s', define ht(w ) = ~ - l h ( w ) . B y 7.1, hi is continuous a t each p o i n t of ~ - s ' ; since s' is open in ~, h~ is still continuous at each p o i n t of s'. T h e c o m p o n e n t s of ~ - l h ( s ' ) are t h e sets h~(s'); since ~ is compact, their closures are t h e sets h~ (~). Finally, g is a h o m e o m o r p h i s m of hi (~) o n t o h (~), since hi (s) is compact.
6 - 563804. Acta mathematica. 97. I m p r i m 6 le 11 a v r i l 1957.
82 JAMES MUNKRES
7.4. LEMMA. Let s be a simplex; let h map ~ homeomorphieally into X , mapping s into X - A. Then there is a unique lilting o / h to a homcomorphism h* o / ~ into X*.
Proo/. L e t s' d e n o t e t h e set ~ - h -1 (A). I f z Es', we m u s t define h* (z) = :~-lh
(Z).
L e t zE~ - s', a n d let x = h(z). Choose a p o l y h e d r a l n e i g h b o r h o o d (St t, k, J) of x such t h a t d i m t = ]. There is one a n d only one c o m p o n e n t C of St t - t such t h a t C N h(s') has x as a limit point: Since x is a l i m i t point of (St t N h(s')) c (St t - t), there is at least one such c o m p o n e n t C. On t h e other hand, if U~ is the e-neighborhood of z in ~ (where e is small e n o u g h t h a t h(U~) c St t), t h e n h(U~ N s'), being connected, is contained in some c o m p o n e n t C of St t - t. Hence there is o n l y one such c o m p o n e n t C.
Define h*(z) = w(C, x). I f h* is to be continuous, h*(z) m u s t be so defined. F o r h*(z) m u s t be a limit p o i n t of h*(U~ N s ' ) ~ 7~-1(C)c W(C), a n d co(C, x) is o n l y t h e point of
~-1(x) which can be a limit p o i n t of W(C).
This definition of h* (z) is i n d e p e n d e n t of the choice of (St t, k, J). L e t (St t', k', J') be a p o l y h e d r a l n e i g h b o r h o o d of x such t h a t dim t' = ?'. x is a limit p o i n t of (St t) N (St t') N h(s'), so t h a t x is a limit point of C N C' N h(s'), for some c o m p o n e n t C' of St t' - t'. I f (St t', k', J') were used to define h* (z), we would define h*(z) = eo(C', x). B u t o~ (C', x) = o~ (C, x).
Finally, we show t h a t h* is continuous at z. L e t W ( D ) be a n e i g h b o r h o o d of h*(z) of t y p e (B), derived f r o m (Str, g, L). T h e n ~o (D, x) = h* (z). L e t e be small e n o u g h t h a t h ( U ~ ) c Str. W e show t h a t h*(U~)a W(D). First let z ' E U ~ - s ' . N o w h ( U ~ N s ' ) c D, so t h a t D is t h e o n l y c o m p o n e n t of S t r - r such t h a t z' could be a limit p o i n t of D N h (s'). B y t h e preceding p a r a g r a p h , we m u s t h a v e h* (z') = co (D, h (z)) ~ W (D). H e n c e h* (U, - s') ~ W (D). Since also h* ( Us N s') c g-1 (D) a W (D), it follows t h a t h* (U~) ~ W (D).
h* is a u t o m a t i c a l l y continuous at each p o i n t of s', so t h a t it is continuous on ~. I t is
1 - 1, since h is, so t h a t it is a h o m e o m o r p h i s m on ~. W e h a v e a l r e a d y shown t h a t h* is unique.
7.5. DEI~II~ITION. L e t K ' be a subdivision of K such t h a t for each simplex s of K ' , t h e simplex of K c o n t a i n i n g s contains a v e r t e x of s. T h e n K ' is called an admissible sub- division. E x a m p l e s of such subdivisions include t h e b a r y c e n t r i c subdivisions of K a n d t h e
" m o d i f i e d " baryeentric subdivisions defined in 3.5.
7.6. LEMMA. Let / be a map o/the complex K into X; consider the/ollowing conditions on the pair (K, /):
(1) Both/-1 (-~k) a n d / - x (Ak) contain every simplex o / K which they intersect,/or each k.
(2) I / t h e vertices o / a simplex o / K lie i n / - 1 (~k), so does the simplex.
(3) I / a simplex lies i n / - 1 (Ae), so does at least one o / i t s vertices.
T H E T R I A N G U L A T I O N O F L O C A L L Y T R I A N G U L A B L E S P A C E S 83 Let ( K , / ) satis/y condition (1), and let K ' be an admissible subdivision o/ K. Then (K', /) satis/ies all three conditions.
7.7. LEMMA. Let h be a homeomorphism o I K into X such that (K, h) satis/ies conditions (1) to (3) o I 7.6. Then there is a triangulation ( L, g) o I 7~-lh ( I K I ) and a strongly linear map p o / L onto K such that hp =7~g (see 2.1).
Proo/. W e d e f i n e a n a b s t r a c t c o m p l e x J as follows: L e t V d e n o t e t h e set of v e r t i c e s of K . L e t W be a set which is in o n e - t o - o n e c o r r e s p o n d e n c e w i t h t h e set g - l h ( V ) ; if y E z - l h ( V ) , l e t y' d e n o t e t h e c o r r e s p o n d i n g p o i n t of W. T h e v e r t i c e s of J a r e t h e p o i n t s of W. T h e s i m p l i c e s of J a r e d e f i n e d as follows:
(a) L e t s = v 0 ... vm b e a s i m p l e x of K l y i n g in h - l ( A k ) . T h e n ~-lh(s) consists of /r d i s j o i n t h o m e o m o r p h s of s, b y 7.3. L e t t b e one of t h e c o m p o n e n t s of ~ - l h (s); t h e n i c o n t a i n s
! !
p r e c i s e l y one p o i n t y~ of t h e set ze-lh(v~) for e a c h i. L e t {Y0 . . . ym} be a s i m p l e x of J . T h e r e a r e k such d i s t i n c t simplices, b y c o n d i t i o n (3).
(b) L e t s = v 0 ... vm b e a s i m p l e x of K l y i n g in h -1 ( X - A). T h e r e is a u n i q u e l i f t i n g o f t h e m a p h of ~ i n t o X i n t o a m a p h* of ~ i n t o X*. h* (~) c o n t a i n s e x a c t l y one p o i n t y~ o f
! !
t h e set 7e-lh(v~), for e a c h i. L e t {Y0 . . . . , Ym} b e a s i m p l e x in J . E a c h such s i m p l e x is d i s t i n c t f r o m one of t h e p r e v i o u s t y p e (if t h e v e r t i c e s of a s i m p l e x s lie in h -1 (Ak) a n d s o m e v e r t e x lies in h -1 (Ak), t h e n s c h -~ (Ak)).
E a c h face of a s i m p l e x of J is also a s i m p l e x of J (the f a c t t h a t h* is u n i q u e is e s s e n t i a l here). T h e r e is a n a t u r a l p r o j e c t i o n of J o n t o K w h i c h carries y' i n t o h-17e(y); i t is s t r o n g l y l i n e a r a n d m a p s a t m o s t a f i n i t e n u m b e r of s i m p l i c e s of J o n t o a n y one s i m p l e x of K . H e n c e J is a l o c a l l y f i n i t e a b s t r a c t c o m p l e x .
L e t L b e a g e o m e t r i c r e a l i z a t i o n of J ; l e t y* d e n o t e t h e v e r t e x of L c o r r e s p o n d i n g t o t h e v e r t e x y ' of J . L e t p b e t h e l i n e a r e x t e n s i o n t o L of t h e m a p w h i c h carries t h e v e r t e x y* i n t o h-lg(y). W e d e f i n e t h e h o m e o m o r p h i s m g. L e t s = y ~ . . . y * b e a s i m p l e x of L . I f p(s) lies in h - l ( A k ) , t h e n ~-lhp(s) consists of /r c o m p o n e n t s , t o one of whose closures Y0 . . . y~ belong. ~ is a h o m e o m o r p h i s m of t h i s closure o n t o hp(~), so t h a t z - l h p = g m a p s ~ h o m e o m o r p h i c a l l y o n t o t h i s closure. On t h e o t h e r h a n d , if p(s) lies i n h -1 (X - A ) , t h e r e is a u n i q u e l i f t i n g of t h e m a p h of p (~) i n t o X t o a m a p h* of p (~) i n t o X*. L e t g = h* p
o n 8 .
T h e m a p g is d e f i n e d on e a c h closed s i m p l e x of L; i t is r e a d i l y v e r i f i e d t h a t t h e s e d e f i n i - tions a g r e e o n c o m m o n faces, so t h a t g is c o n t i n u o u s . I t is also r e a d i l y v e r i f i e d t h a t g i s 1 - 1, t h a t i t is a l i f t i n g of hp, a n d t h a t i t m a p s [L] onto ~ - l h ( ] g [ ) .
I t follows t h a t g is a h o m e o m o r p h i s m : L e t v b e a v e r t e x of K . T h e m a p g is a h o m e o - m o r p h i s m on Cl (St (p-1 (v))), since t h i s closure is c o m p a c t . H e n c e i t suffices t o s h o w t h a t
84 JAMES MUNKRES
g (St (p-1 (v))) is open in g ( [L t )" B u t this set equals 7t; - l h (St v), which is open in ~-~ h ( [ K ] ) = g([L]).
7.8. P R O r O S I T I O N . X* is a locally polyhedral space o/ dimension n; it has a naturally induced locally polyhedral structure relative to which z preserves index and X* =
Cl (X* - (X*) n-~).
Proo/. We define a basis B for a locally polyhedral structure on X*. L e t (U, k, J ) be a polyhedral neighborhood on X; let V be an open set such t h a t V c U and ] J [ - k (V) is a polyhedron in J . Consider a subdivision J' of J in which I J I - k (V) is a subcomplex;
then k(V) is a subcomplex H of J ' . The pair (H, k - l t H ) satisfies condition (1) of 7.6. Let H' be a n y admissible subdivision of H. Then there is a triangulation (L, g) of ~ - 1 ( ~ ) and a strongly linear m a p p of L onto H ' such t h a t k - l p =~g. The set z - l ( V ) is open in X*, a n d C l ( g - l ~ - ~ ( V ) ) = I L l . Moreover, I n [ - g - l ~ - l ( V ) is a subcomplex of L, because
l H ' [ - k (V) is a subcomplex of H ' . Define
(:Tg -1 (V), g - l , L)
to be an element of B. This polyhedral neighborhood on X* is said to be derived [rom (U, k, J). Obviously, B covers X*.
L e t yEX*; let x -7r(y). Then I s ( y ) < I(x), since each element of B is derived from a polyhedral neighborhood on X. L e t (U, k, J ) be a polyhedral neighborhood of x. I f k(x) lies in an m-simplex of J , there is a complex H ' defined as in the preceding p a r a g r a p h such t h a t k(x) lies in an m-simplex of H ' (by3.5). Then y lies in an m-simplex of the derived neighborhood (~z -1 (V),
g-i, L).
Hence I s (y) > I (x).Since I B (y) = I(x), I B is constant on each simplex of an element of B. Hence B is a basis for a structure, and relative to this structure, 7r preserves index.
To show t h a t X* is metrizable, note t h a t it is a Hausdorff space with a countable basis;
being locally compact, it is also regular. E v e r y regular space with a countable basis is metrizable ([1], p. 81). Hence X* is a locally polyhedral space. X * = C l ( X * - ( X * ) n - 1 ) , since every neighborhood of t y p e (A) or (B) (see 6.2) m u s t intersect ~ - I ( X - X n - 1 ) .
7.9. PROPOSITION. X* has no singular points o/ index greater than j -- 1.
Proo[. Let y be a point of X* having index greater t h a n j - 1; let x = ~ ( y ) . First, suppose x E X - ~ . Then x is non-singular. L e t V be a neighborhood of y such t h a t V is compact and does not intersect rr -1 (~). Then Jr is a homeomorphism on V. Since V = 7r -1 (~r (V)) and ~r is strongly continuous, rr (V) is open in X. Now rr maps V homeomorphically
onto 7r(V) and carries (X*) n-1 onto X n-l, so t h a t x and y have the same singularity (by 5.1). Hence y is non-singular.
T H E T R I A N G U L A T I O : N O F L O C A L L Y T R I A N G U L A B L E S P A C E S 85 Second, suppose x C-~. T h e n x E Ak for some k, since A - 13 Ak; a n d I ( x ) - ] . Some n e i g h b o r h o o d U of x in X j consists entirely of points h a v i n g t h e same singularity a n d index as x (4.3); since U intersects Ak, xEAk. L e t W(C) be a n e i g h b o r h o o d of y of t y p e (B), derived f r o m the polyhedral n e i g h b o r h o o d (Sts, h, L) of x. N o w z maps W(C) homeo- m o r p h i c a l l y onto C tJ s, m a p p i n g W(C) - (X*) j onto (C U s) - X s = C, which is connected.
I n the preceding p a r a g r a p h , we showed t h a t X* has no singular points of index greater t h a n ?" (since A c XJ). T h e n b y 5.7 a n d 6.4, y is non-singular.
8. T r i a n g u l a t i n g t h e s p a c e . The following two definitions hold for a n y locally poly- hedral space X, w i t h o u t the a s s u m p t i o n s t a t e d at the beginning of this chapter.
8.1. DEFINITIOr L e t (K, h) be a t r i a n g u l a t i o n of some subset of X, a n d let (U, k, J) be a p o l y h e d r a l n e i g h b o r h o o d of x. T h e n (K, h) a n d ( U, k, J) are said to be locally compatible at x if there is a n e i g h b o r h o o d V of h -1 (x) in I K[ such t h a t (1) h ( V ) c U, a n d (2)there is a subcomplex H of some subdivision of K such t h a t I HI - V a n d kh is a linear m a p of H into J . The set kh(V) is clearly a p o l y h e d r o n in J ; it follows f r o m 1.1 t h a t there are sub- divisions H ' a n d J ' of H a n d J respectively, such t h a t kh a linear isomorphism of H ' onto a subcomplex of J ' .
8.2. D E F I N I T I O N . The locally p o l y h e d r a l space X is said to possess property T if t h e following holds:
L e t t h e points of X ~ be d e n o t e d b y Xl, x2,... Given a p o l y h e d r a l n e i g h b o r h o o d (Ui, k~, J~) of x~, for each xi, a n d a proper triangulation of X ~ 1 which is locally compatible with (Us, ]c~, J~) at xi, there exists a proper t r i a n g u l a t i o n of X which is compatible with the given t r i a n g u l a t i o n of X ~ 1 a n d locally compatible with (U~,/c~, J~) at x~, for each i.
(The purpose of the "local c o m p a t i b i l i t y " p a r t of this definition will become clear l a t e r - - 9.11.)
8.3. LEMMA. Let p be a strongly linear map o/the complex L onto K; let p-1 (x) be/inite /or each x. I / L ' is a subdivision o/ L, there are subdivisions L" and K" o/ L' and K respectively, such that p is a strongly linear map o / L " onto K".
Proo/. If K" is a n y subdivision of K, t h e n K" n a t u r a l l y induces a subdivision L" of L such t h a t p is a strongly linear m a p of L" onto K". We m u s t so choose the subdivision K"
t h a t the induced subdivision L" is a subdivision of L'. W e proceed as follows: F o r each simplex s of K, there are a finite n u m b e r of simplices t 1 .. . . . t k of L which p m a p s homeo- m o r p h i c a l l y onto s. L' induces a subdivision of each set ti; t h e n L' induces several subdivi- sions of ~. Order t h e simplices of K: el, s2, ... L e t K 0 = K; suppose a subdivision Kin_ 1 of K is given. N o w L ' induces several subdivisions of sin, a n d Kin_ 1 induces a n o t h e r sub-
