A combinatoric invariant of simple polytopes 1 Simple polytopes 2 - - PowerPoint PPT Presentation

▶
a combinatoric invariant of simple polytopes
SMART_READER_LITE
LIVE PREVIEW

A combinatoric invariant of simple polytopes 1 Simple polytopes 2 - - PowerPoint PPT Presentation

Feb 25, 2024 424 likes •839 views

A combina- toric invariant of simple polytopes Bo Chen A combinatoric invariant of simple polytopes 1 Simple polytopes 2 graded Boolean Bo Chen ring of simple polytopes School of Mathematics and Statistics, HUST 3 Jiont-work

slide-1
SLIDE 1

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

A combinatoric invariant of simple polytopes

Bo Chen

School of Mathematics and Statistics, HUST Jiont-work with Zhi L¨ u and Li Yu bobchen@hust.edu.cn

  • Apr. 2014, SJTU, Shanghai
slide-2
SLIDE 2

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

convex polytope

A convex polytope is defined to be a convex hull of finite many points in Rn.

slide-3
SLIDE 3

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

convex polytope

A convex polytope is defined to be a convex hull of finite many points in Rn. the intersection of finite number of half-spaces of Rn.

slide-4
SLIDE 4

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

face

Definition

Let H be a half-space of Rn. H ∩P is called a face of a convex polytope P, if H ∩ int(P) = ∅. A 0-face is called a vertex, a 1-face is called an edge and a (n − 1)-face is called a facet.

slide-5
SLIDE 5

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

simple convex polytope

Definition

A convex polytope P n is simple if each vertex adjoint to exactly n facets. if each vertex adjoint to exactly n edges.

slide-6
SLIDE 6

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

simple convex polytope

Definition

A convex polytope P n is simple if each vertex adjoint to exactly n facets. if each vertex adjoint to exactly n edges.

slide-7
SLIDE 7

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

simple convex polytope

Definition

A convex polytope P n is simple if each vertex adjoint to exactly n facets. if each vertex adjoint to exactly n edges. if each k-face is the intersection of exactly k facets.

slide-8
SLIDE 8

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

slide-9
SLIDE 9

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

combinatoric of convex polytope

Definition

A face lattice of a convex polytope P is a poset (F, ≤), where F is the set of faces of P, and a ≤ b iff a ⊂ b, a, b ∈ F.

Definition

Two polytopes are called combinatorially isomorphic if their face lattices are isomorphic.

slide-10
SLIDE 10

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

some combinatoric invariants of simple polytopes

f-vector,h-vector

Definition

Let P n be a simple convex polytope, and suppose fi be the number of (n − i − 1)-faces of P n, i = −1, 0, 1, · · · , n − 1.

slide-11
SLIDE 11

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

some combinatoric invariants of simple polytopes

f-vector,h-vector

Definition

Let P n be a simple convex polytope, and suppose fi be the number of (n − i − 1)-faces of P n, i = −1, 0, 1, · · · , n − 1.Let hk =

k

  • i=0

(−1)k−i d − i k − i

  • fi−1.

f(P) = (f−1, f0, f1, · · · , fn−1) is called the f-vector of P, and h(P) = (h0, h1, · · · , hn) is called the h-vector of P. Actually,

n

  • i=0

fi−1(t − 1)n−i =

n

  • k=0

hktn−k.

slide-12
SLIDE 12

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Stanley-Reisner ring

Stanley-Reisner ring Order the facets of P, say {F1, · · · , Fm}.

slide-13
SLIDE 13

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Stanley-Reisner ring

Stanley-Reisner ring Order the facets of P, say {F1, · · · , Fm}. Define a ideal I(P)

  • f k[x1, · · · , xm],

I(P) = xi1 · · · xis|Fi1 ∩ · · · ∩ Fis = ∅. A Stanley-Reisner ring k[P] of a simple polytope P is defined to be k[P] = k[x1, · · · , xm]/I(P).

slide-14
SLIDE 14

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

vertex-face incident vectors

Definition

For each face f of a polytope P, define a vector of face f, ζf : V (P) → Z2, p →

  • 1,

if p ∈ f; 0, if p / ∈ f. ζf0 = (1, 0, 1, 1, 0)

slide-15
SLIDE 15

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

graded Boolean ring of polytopes

Definition

Bk(P)

  • span{ζf | f is a codim-k face of P},

if k ≤ n; V∗ = Map(V (P), Z2) ∼ = Zs

2,

if k > n. B(P)

  • k≥0

Bk tk. ∀ faces f and g of P, define (ζf ◦ ζg)(v) = ζf(v)ζg(v). Then ζf∩g = ζf ◦ ζg. Suppose P n is simple. Each k-face of a simple convex poly- tope is the intersection of exactly k facets, So B(P) becomes a graded ring.

slide-16
SLIDE 16

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Proposition

dim B1(P) ≥ h0 + h1.

Remark

Actually, choose any vertex v of P and n−1 facets incident to v, then face-vectors of remaining facets(of number m−n+1 = h0 + h1) are linearly independent in B1(P).

slide-17
SLIDE 17

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

slide-18
SLIDE 18

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

n-colorable simple polytopes

Definition

A simple polytope P n is n-colorable, if there is a map c : F → [n], such that c(Fi) = c(Fj) if Fi ∩ Fj = ∅.

slide-19
SLIDE 19

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Theorem

The following statements are equivalent. P n is n-colorable.

Remark

The proof of equivalence of the first two statements can be found in [Jos].

slide-20
SLIDE 20

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Theorem

The following statements are equivalent. P n is n-colorable. each 2-face of P has even number of vertices.

Remark

The proof of equivalence of the first two statements can be found in [Jos].

slide-21
SLIDE 21

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Theorem

The following statements are equivalent. P n is n-colorable. each 2-face of P has even number of vertices. B1(P) = m − n + 1.

Remark

The proof of equivalence of the first two statements can be found in [Jos].

slide-22
SLIDE 22

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Theorem

The following statements are equivalent. P n is n-colorable. each 2-face of P has even number of vertices. B1(P) = m − n + 1. B0 ⊂ B1 ⊂ · · · ⊂ Bn.

Remark

The proof of equivalence of the first two statements can be found in [Jos].

slide-23
SLIDE 23

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

graded dimension of the ring

Proposition

Suppose P n is an n-colorable simple polytope. Then dim Bi(P) =

  • h0 + h1 + · · · + hi,

if i ≤ n − 1; ♯(V ert(P)), if i ≥ n.

slide-24
SLIDE 24

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Self-dual codes risen from n-colorable polytopes

Theorem

For any n-colorable simple polytope P n with n odd, there is a binary self-dual code W = B[ n

2 ](P) = span{ζf|f is a n + 1

2

  • face of P}.

Remark

For the case n = 3, a basis of the self-dual code W risen from P 3 can be written down quickly: {ζf|f ∈ F(P 3) \ {f1, f2}} where f1 and f2 are any two faces with a common edge.

slide-25
SLIDE 25

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

self-dual codes [12, 6, 4] risen from 6-prim

                     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                     

12×6

slide-26
SLIDE 26

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

properties of such self-dual codes

Proposition

Let W be a self-dual code realized by an n-colorable simple n- polytope(n is odd). Let W is of type [l, l/2, d]. Then l ≥ 2n. If n = 3, d = 4.

Remark

Different polytopes may give same self-dual code. Take the connected sum of two 6-prim.

Remark

Extended Golay code can not be realized by such a polytope.

slide-27
SLIDE 27

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Further questions

Conjecture: d = min{♯ζf|f is a n+1

2 -face of P}(≥ 2

n+1 2 ),

for any self-dual code W risen from P n.

slide-28
SLIDE 28

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Further questions

Conjecture: d = min{♯ζf|f is a n+1

2 -face of P}(≥ 2

n+1 2 ),

for any self-dual code W risen from P n. dimBk for general simple polytope.

slide-29
SLIDE 29

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Background

Puppe and Kreck ([Puppe][KP])

{Involution on 3-manifolds with isolated fixed pts} ← → {self-dual codes}

slide-30
SLIDE 30

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Background

Puppe and Kreck ([Puppe][KP])

{Involution on 3-manifolds with isolated fixed pts} ← → {self-dual codes} In case of dim=3, the equvi. cohomology are one-to-one corresponding to binary self-dual codes.[KP] For higher dimension case, only the odd dimension case gives self-dual codes.[Puppe][CL]

slide-31
SLIDE 31

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Background

Puppe and Kreck ([Puppe][KP])

{Involution on 3-manifolds with isolated fixed pts} ← → {self-dual codes} In case of dim=3, the equvi. cohomology are one-to-one corresponding to binary self-dual codes.[KP] For higher dimension case, only the odd dimension case gives self-dual codes.[Puppe][CL] H

[ n

2 ]

Z2 (Mn) becomes a self-dual code.

slide-32
SLIDE 32

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Background

Puppe and Kreck ([Puppe][KP])

{Involution on 3-manifolds with isolated fixed pts} ← → {self-dual codes} In case of dim=3, the equvi. cohomology are one-to-one corresponding to binary self-dual codes.[KP] For higher dimension case, only the odd dimension case gives self-dual codes.[Puppe][CL] H

[ n

2 ]

Z2 (Mn) becomes a self-dual code.

[CL] gives a lower bound for number of self-dual codes.

slide-33
SLIDE 33

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Motivation

What can we do on the categary of small covers?

slide-34
SLIDE 34

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Motivation

What can we do on the categary of small covers? Roughly speaking, a small cover is a closed manifold deter- mined by a simple polytope with coloring. Let M(P n, λ) be a small cover over (P, λ), then there is a natural Zn

2-action.

Moreover we proved that

Lemma

M(P n, λ) admits a rank 1 sub-action(i.e., a Z2-action) with isolated fixed points ⇐ ⇒ (P, λ) is n-colored.

slide-35
SLIDE 35

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

{small covers}

PuppeKreck

  • {self-dual codes}

{simple polytopes}

?

  • DJ
slide-36
SLIDE 36

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Let P n be n-colored, then Z2[P]

∼ = H∗ Zn

2 (M)

φ∗

  • i∗
  • g
  • H∗

Z2(M) i∗

  • H∗

Zn

2 (V )

φ∗

|V

H∗

Z2(V )

Theorem

i∗(H∗

Z2(M)) = B(P), where i∗ is injective.

slide-37
SLIDE 37

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Let P n be n-colored, then Z2[P]

∼ = H∗ Zn

2 (M)

φ∗

  • i∗
  • g
  • H∗

Z2(M) i∗

  • H∗

Zn

2 (V )

φ∗

|V

H∗

Z2(V )

Theorem

i∗(H∗

Z2(M)) = B(P), where i∗ is injective.

Corollary

B(P) ∼ = Z2[P]/I, where I is generated by

  • F∈F(α1)

xF +

  • F∈F(αi)

xF , i = 2, 3, · · · , n.

slide-38
SLIDE 38

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Reference

  • C. Allday, V. Puppe,

Cohomological methods in trans- formation groups. In: Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge University Press, Lon- don (1993).

  • B. Chen, Z. L¨

u, Equivariant cohomology and analytic de- scriptions of ring isomorphisms, Math. Z. 261 (2009), No. 4, 891–908.

  • M. Joswig, Projectivities in simplicial complexes and col-
  • rings of simple polytopes, Math. Z. 240 (2002), no. 2,

243–259.

  • M. Kreck and V. Puppe, Involutions on 3-manifolds and

self-dual, binary codes, Homology, Homotopy Appl. 10 (2008), no. 2, 139–148.

  • V. Puppe, Group actions and codes. Can. J. Math. l53,

212–224 (2001).

slide-39
SLIDE 39

A combina- toric invariant of simple polytopes Bo Chen §1 Simple polytopes §2 graded Boolean ring of simple polytopes §3 n-colorable simple polytopes §4 Binary self-dual codes §5 Motivation Reference §0 Thanks

Thanks for your attention!