Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/TTC/slides/gaifullin.pdf Äàòà èçìåíåíèÿ: Fri May 26 08:51:30 2006 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:27:00 2016 Êîäèðîâêà: Windows-1251

Euler characteristic. (K ) = f0(K ) - f1(K ) + f2(K ) - . . . (K 2 ) =
v K 2

dv 1- 6

,

where dv is the number of edges entering v .

Stiefel-Whitney classes. W n (K ) =
nK

n

(mod 2).

Theorem (Whitney, 1940, Halperin, Toledo, 1972) [Wn(K )] is the Poincar? dual of wm-n(K ), e where m = dim K .

1


Rational Pontrjagin classes. Rokhlin, Swartz, Thom, 1957-1958: Rational Pontrjagin classes are well defined for combinatorial manifolds. Problem. Given a combinatorial manifold K construct explicitly a rational simplicial cycle Z (K) representing the Poincar? dual of pk (K). e


Formulae. ž Gabrielov, Gelfand, Losik, 1975, MacPherson, 1977. A formula for the first Pontrjagin class of any Brouwer manifold. ž Cheeger, 1983. Formulae for all Pontrjagin classes. - Include calculation of the spectrum of the Laplace operator. - Give only real cycles. ž Gelfand, MacPherson, 1992. Formulae for all Pontrjagin classes of a triangulated manifold with given smoothing or combinatorial differential (CD) structure. - Do not solve the above problem.


Lo cal formulae. link = { K | K, = } . f (K m ) =
m-nK m

f (link ).

f is a skew-symmetric rational-valued function on the set of isomorphism classes of oriented (n - 1)-dimensional PL spheres. f does not depend on K . Problem. Describe all functions f such that f (K ) is a cycle for every K . f if P is [f (p a local formula for P Q[p1, p2, . . .] (K )] is the Poincar? dual of e 1(K ), p2(K ), . . .) for every K .


Bistellar moves. Theorem (Pachner, 1989). Two combinatorial manifolds are PL homeomorphic iff the first can be transformed into the second by a finite sequence of bistellar moves.

r r r ' E

r r r r

r d d d r d d d d d d d d dr d d d d d r ' E r d d d d

r d d d d d d d d dr d d dr

d

d d


r ? ? ? ? ? ? ? ? ? ? ? r ? d ? d ? d $r $$$ ? d $$ $ d ? $$$$ d?$ r

'

E

r ? ? ? ? ? ? ? ? ? ? ? r r ? d ? d ? d $r $$$ ? d $$ $ d ? $$$$ dr $ ?

r ? ? ? ? ? ? ? ? ? ? ? r ? d e ? ed ? ed $r ? $$$ ed $$ $ e d ? $$$$ dr ?$ e g e g e g e eg eg eg eg eg eg eg gg er

'

E

r ? ? ? ? ? ? ? ? ? ? ? r ? d e ? ed ? ed $r ? $$$ ed $$ $ e d ? $$$$ dr $ ? e g e g e g e eg eg eg eg eg eg eg gg er


Lo cal formulae for the first Pontrjagin class. oriented 3-dim. rational f: PL-sphere L number f (L) - - - L = L1 - - L2 - - ... - - 4
bistellar moves
1 2 q

- Lj - - Lj +1, link
j,v
Lj

j

v is a vertex of Lj
Lj +1

v - - link --

v

Graph 2. Vertices: isomorphism classes of oriented 2-dimensional PL spheres. Edges: bistellar moves.
q

=
j =1 v Lj

j,v C1(2; Z)

f (L) = c( ), c C 1(2; Q).


Theorem (G., 2004) There is a cohomology class c H 1(2; Q) such that local formulae for the first Pontrjagin class are in one-to-one correspondence with cocycles c C 1(2; Q) representing c. The correspondence is given by the formula f (L ) = c ( ) Cohomology class c. The group H1(2; Z). Generators: 6 infinite series. Let us give the values of c on these generators.

q-p (p, q ) = (p + q + 2)(p + q + 3)(p + q + 4) 1 (p ) = (p + 2)(p + 3)


s t t t t t ts t s

s t t t t t ts t s

s t t t E t Ås Årr t Å rr t t rs sÅ Å

s t t t t t ts t s

T s t t t t t ts t s s t t t ' s År t t ÅÅ rr Å t rs t r Å s

0

s t t t t Ås Årr t Å rr t t rs Å sÅ

c

s t t t t Ås r t ÅÅ rr Å t rs t r Å s

s 3s 33 33 s 33 3 3 33 3 33 s s 33

p triangles q triangles
T

E

s 3s e 33 33 e 33 es e s 33 ? 33 ? 3 ? 33 s ?3 3 s

(p, q )
' s c 3s e 33 ? 33 e ? 33 es ? 3 e ? s s 33 ? e ? 33 e ? 33 e ?3 3 s s e

s 3s 33 ? 33 ? s 33 ? ? 3 s 33 e 33 e 33 e 3 s3 s e

p triangles
3s 33 3 33 s s3 3 33 33 3 s 33 3 E 3s ? 33 ? 3 ? 33 s ? s3 s 3 33 e 33 e 33 s 3 e3 e

q triangles
T

(0, q ) - (0, p)
' c 3s ?e 33 ? e 3 ?e 33 s 3 s3 ? s es 33 e ? 3 e ? 333 s 3 e3 ? e?

3s 33 e e 3 e 33 s 3 s3 es 33 ? 33 ? 3 s 33 ? ? 3


p triangles

s s Å Å rr Å rr Å Å r sÅ r s Å ¿ ¿

q triangles Å

r triangles

s f ? ? f ? f ? f ? f fs s ? ' Å ÅÅ d Å d ÅÅ d Å ds s Å

U

( p ) - (q ) +(r) -
1 12

w

s ?f ? f ? f ? f ? f fs s ? r rr d rr d rr d ds r s

q triangles r Å gles trian Å
s s ¿ s ¿ Å s ¿ E

s

p triangles k triangles T
s d d d d d s d ds d ' s

s s Å Å rr Å r sr r Å s rr j r

(p ) - ( q ) - (r ) + (k )
s s ÅÅ Å d t % Å d t td td td ts Åd Å rr Å d rs d r s Å

s s t t t t t t s Årr ÅÅ rs r s Å

q triangles p triangles

s d ?? ?d dÅ ? ds s ? e ? ¿ e ? ? ? e? e? s ? s ¿ ¿ T

r triangles Å

k triangles
E

l triangles

s d d d ds s e ? ? e ? e rr es ? s r j r

(p ) + ( q ) + (r ) +(k ) + (l) -
Å s ÅÅ g d % Å gd gd ds s g e ? g ? e g? e g es g? s

s ?d g ?g d ?g d ? ds ' s g ? e ? g ? e? g? e? gg? e? s s

1 12

s d d d ds s e ? ? e e ? es ? s


The cochain complex T (Q). T n(Q) is the vector space of all skew-symmetric rational-valued functions on the set of isomorphism classes of oriented (n - 1)-dimensional PL spheres. : T n(Q) T n+1(Q); ( f )(L) =
v L

f (link v );

2 = 0.

f (K ) is a cycle for every K f is a cocycle. f (K ) is a boundary for every K f is a coboundary.


Existence and uniqueness. ž H (T (Q)) Q[p1, p2, . . .], deg pi = 4i. = ž Each cocycle of T (Q) is a local formula for some polynomial in rational Pontrjagin classes. ž A local formula for a polynomial in rational Pontrjagin classes exists and is unique up to a coboundary. (The existence strengthens a result of Levitt and Rourke, 1978.) ž We describe explicitly the cohomology class H 4(T (Q)) such that () = p1. ž We describe explicitly the cohomology classes i H 4i(T (Q)) such that (i) = Li(p1, . . . , pi).


Denominators. For f T n(Q), by denl (f ) we denote the least common multiple of the denominators of the values f (L), where L runs over all (n - 1)-dimensional oriented PL spheres with not more than l vertices. ž H (T (Q)) there exist a cocycle f representing and an integer constant C such that denl (f ) is a divisor of C (l + 1)! for any l. ž Suppose f is a local formula for the first Pontrjagin class. Then denl (f ) is divisible by the least common multiple of the numbers 1, 2, . . . , l - 3 for any even l 10. ž H 4(T (G)) = 0 for any subgroup G Q. Recall that H 4(T (Q)) = Q.