By J. F. Davis

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**Example text**

As a subspace of is c l e a r , L b L C C~LC VL. Let EL = TV Ib • Next, c o n s i d e r a s i m p l e x ~ of L L b L (which corresponds to a s i m p l e x of ZL). Abusing our e s t a b l i s h e d n o t a t i o n m i n i m a l l y , we o b t a i n a face h(L,~): b L of L and a map which i d e n t i f i e s b w i t h the dual c e l l to o in L L b . h(L,o) extends to a map~ ~: V + Rn+k That i s , L L o n+k t h i n k of V as Q (~) X Then h ( L , o ) extends to ¢: Q + R L L L L + , ~R, by @: tp + ~ ( L , ~ ) ( O ) ~ t ( h ( L ~ o ) ( p ) - h ( L , o ) ( O ) ) .

E . q, in M = @W. ) some Under t h i s neighborhood of M. the s i g n a t u r e of the o r i e n t e d m a n i f o l d There are then two is merely the usual a l g e b r a i c - t o p o l o g i c a l s i g n a t u r e of a m a n i f o l d - w i t h - b o u n d a r y i . e . form in (If We assume W be equipped w i t h a Riemannian m e t r i c i s o m e t r i c obvious " c a n d i d a t e s " f o r The f i r s t 4i-i M o r i e n t e d Riemannian ( 4 i - 1 ) - m a n i f o l d s 2J-dimensional r a t i o n a l the s i g n a t u r e of homology.

To define EL, is a simplex of we f i r s t ZL" ~L,o' where We may think of these as subspaces of C~L, w i l l r e s t r i c t to a homeomorphism on L T a k e the second b a r y c e n t r i c s u b d i v i s i o n of C~L ( n o t i n g t h a t the f i r s t first and CZL ÷ e since the n a t u r a l map these spaces• define spaces EL, * s u b d i v i s i o n of s u b d i v i s i o n is the s i m p l i c i a l cone on the ZL, a s i m p l i c i a l complex). which we w r i t e as i cZ , to specify i t Call t h i s second s u b d i v i s i o n , ( i .