Euler characteristic. ( K ) = f 0 ( K ) f 1 ( K ) + f 2 ( K ) . . - PDF document

Euler characteristic. χ ( K ) = f 0 ( K ) − f 1 ( K ) + f 2 ( K ) − . . . � � � 1 − d v χ ( K 2 ) = , 6 v ∈ K 2 where d v is the number of edges entering v . Stiefel-Whitney classes. � σ n W n ( K ) = (mod 2) . σ n ∈ K ′ Theorem ( Whitney,1940, Halperin,Toledo,1972 ) [ W n ( K )] is the Poincar´ e dual of w m − n ( K ), 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 combinatorialmanifold K construct explicitly a rational simplicial cycle Z ( K ) representing the Poincar´ e dual of p k ( K ).

Formulae. • Gabrielov, Gelfand, Losik, 1975, MacPherson, 1977. A formula for the first Pontrjagin class of any Brouwer manifold . • Cheeger, 1983. Formulae for all Pontrjagin classes. – Include calculation of the spectrum of the Laplace operator. – Give only real cycles. • 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.

Local formulae. link σ = { τ ∈ K | σ ∪ τ ∈ K, σ ∩ τ = ∅} . � f ♯ ( K m ) = f (link σ ) σ. σ m − n ∈ K m 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 is a local formula for P ∈ Q [ p 1 , p 2 , . . . ] if [ f ♯ ( K )] is the Poincar´ e dual of P ( p 1 ( K ) , p 2 ( 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 ❚ ✑◗◗◗◗◗◗◗◗ ✔ ✔ ✑✑✑✑✑✑✑✑ ❚ ❚ ✔ ✔ ❚ ❚ ✔ ✔ ❚ ❚ ✔ ✔ ❚ ❚ ✔ ✔ ❚ ❚ ✔ ❚ ✔ ◗ ❚ r r r r r r �❅ �❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � ❅ ✛ ✲ � ❅ r r r r ❅ � ❅ � � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ � � ❅ ❅ ❅� ❅� r r

r r ✓ ✆❚ ✓ ✆❚ ✆ ✆ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✛ ✲ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ r r r ✆ ❚ ✆ ❚ ❅ ❅ ✆ ❚ ✆ ❚ ❅ ❅ ✆ ❚ ✆ ❚ ❚ ❚ ❅ ❅ ✘ ✘ ❅✘✘✘✘✘✘✘✘✘✘✘ ❅✘✘✘✘✘✘✘✘✘✘✘ r r ✆ ✆ ❅ ❅ ✆ ✆ ❅ ❅ ✆ ✆ r r r r ✓ ✆ ✆❚ ✓ ✆❚ ✆ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ✆ ❚ ✆ ❚ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ ❚ ❚ ✆ ✆ ✓ ✓ r r ❚ ❚ ✆ ✆ ❆ ❅ ❅ ❆ ❚ ❚ ✆ ✆ ❆ ❅ ❆ ❅ ❚ ❚ ✆ ✆ ❚ ✛ ✲ ❚ ❆ ❅ ❆ ❅ ✘ ✘ ❅✘✘✘✘✘✘✘✘✘✘✘ r ❅✘✘✘✘✘✘✘✘✘✘✘ r ✆ ✆ ✔ ✔ ❆ ❅ ❆ ❅ ✔ ✔ ✆ ✆ ❆ ❅ ❆ ❅ ✔ ✔ ✆ ✆ ❆ ❆ r ✔ r ✔ ❈ ❈ ❆ ❆ ✔ ✔ ❈ ❈ ❆ ❆ ✔ ✔ ❈ ❈ ❆ ❆ ✔ ✔ ❈ ❈ ❆ ❆ ✔ ✔ ❈ ❈ ❆ ✔ ❆ ✔ ❈ ❈ ❆ ✔ ❆ ✔ ❈ ❈ ❆ ✔ ❆ ✔ ❈ ❈ ❆ ✔ ❆ ✔ ❈ ❈ ❆ ✔ ❆ ✔ ❈ ❈ ❆ ✔ ❆ ✔ ❈ ❈ ❆ ❆ ❈✔ ❆ ❆ ❈✔ r r

Local formulae for the first Pontrjagin class. f : oriented 3-dim. rational �→ PL-sphere L number f ( L ) β 1 β 2 βq ∂ ∆ 4 − − − → L 2 − − − → ... − − − → L = L 1 bistellar moves β j − − − → L j +1 , v is a vertex of L j L j β j,v − − − − → link Lj +1 v link Lj v Graph Γ 2 . Vertices: isomorphism classes of oriented 2-dimensional PL spheres. Edges: bistellar moves. q � � γ = β j,v ∈ C 1 (Γ 2 ; Z ) j =1 v ∈ L j c ∈ C 1 (Γ 2 ; Q ) . f ( L ) = � c ( γ ) , �

Theorem (G., 2004) There is a cohomology class c ∈ H 1 (Γ 2 ; Q ) such that local formulae for the first Pontrjagin class are in one-to-one correspondence with c ∈ C 1 (Γ 2 ; Q ) representing c . The cocycles � correspondence is given by the formula f ( L ) = � c ( γ ) Cohomology class c . The group H 1 (Γ 2 ; Z ). Generators: 6 infinite series. Let us give the values of c on these generators. q − p ρ ( p, q ) = ( p + q + 2)( p + q + 3)( p + q + 4) 1 λ ( p ) = ( p + 2)( p + 3)

s s s s ✡❏ ✡❏ ✡❏ ✡❏ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ✲ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ s ✟❍❍❍❍ ✟✟✟✟ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ✡ ❏ ✡ ❏ ✡ ❍ ❏ ✡ ❏ s s s s s s s s ✻ 0 ❄ s s s s ✡❏ ✡❏ ✡❏ ✡❏ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ✡ ✡ ✛ ✡ ✡ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ s s s ✟❍❍❍❍ ✟❍❍❍❍ ✟❍❍❍❍ ✟✟✟✟ ✟✟✟✟ ✟✟✟✟ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ✡ ❏ ✡ ❍ ❏ ✡ ❍ ❏ ✡ ❍ ❏ s s s s s s s s p triangles s s s s ❛❛❛❛❛❛❛❛ ✦ ❛❛❛❛❛❛❛❛ ✦ ✦ ✦ ❆ ✦ ✦ ✦ ✦ ✦ ❆ ✦ ✦ ✦ ✦ ✦ ❆ ❛ ✦ ✦ ❆ ❛ ✦ ✦ ✲ s s s ✦✦✦✦✦✦✦✦ ✦ ❛ ❛ ✦✦✦✦✦✦✦✦ ✦ ❛ ❛ ❛ ❛ ✁ ❛ ✁ ❛ ❛ ❛ ❛ ❛ ✁ ❛ ❛ q triangles ❛ ❛ ❛ ✁ ❛ s s s s ✻ ρ ( p, q ) ❄ ❛❛❛❛❛❛❛❛ s ✦ s ❛❛❛❛❛❛❛❛ s ✦ s ✦ ✦ ✁ ❆ ✁ ✦ ✦ ✦ ✦ ✦ ✁ ❆ ✦ ✁ ✦ ✦ ✦ ✦ ✁ ❆ ✁ ❛ ✦ ✦ ✁ ❆ ❛ ✦ ✦ ✁ ✛ ✦✦✦✦✦✦✦✦ ✦ ❛ ❛ s s ✦✦✦✦✦✦✦✦ s ✦ ❛ s ❛ s ❛ ❛ ❆ ✁ ❆ ❛ ❆ ✁ ❛ ❆ ❛ ❛ ❛ ❛ ❆ ✁ ❆ ❛ ❛ ❛ ❛ ❛ ❆ ✁ ❛ ❆ s s s s p triangles ✦✦✦✦✦✦✦✦ ✦ ❛ ❛ s ✦✦✦✦✦✦✦✦ ✦ s ❛ ❛ ❛ ❛ ✁ ❛ ✁ ❛ ❛ ❛ ❛ ❛ ✁ ❛ ❛ ❛ ❛ ❛ ✁ ❛ ✲ s ❛❛❛❛❛❛❛❛ ✦ s ❛❛❛❛❛❛❛❛ s s ✦ s ✦ ✦ ❆ ✦ ✦ ✦ ✦ ✦ ❆ ✦ ✦ ✦ ✦ ✦ ❆ ❛ ✦ ✦ ❛ ❆ ✦ ✦ s s q triangles ρ (0 , q ) − ρ (0 , p ) ✻ ❄ ✦✦✦✦✦✦✦✦ ✦ ❛ s ❛ ✦✦✦✦✦✦✦✦ ✦ ❛ s ❛ ❛ ❛ ❆ ✁ ❆ ❆ ❛ ✁ ❆ ❛ ❛ ❛ ❛ ❛ ❆ ✁ ❆ ❛ ❛ ❛ ❛ ❆ ❛ ✁ ❆ ❛ ✛ ❛❛❛❛❛❛❛❛ s s ✦ s s ❛❛❛❛❛❛❛❛ s s ✦ s ✦ ✦ ✁ ❆ ✁ ✦ ✦ ✦ ✦ ✁ ✦ ❆ ✁ ✦ ✦ ✦ ✦ ✦ ✁ ❆ ✁ ❛ ✦ ✦ ✁ ❛ ❆ ✁ ✦ ✦ s s