De Rham Cohomology of Finite Abelian Groups

By Lucian Ionescu (Project)

2016

Contents

- Introduction

- What are “Discrete Manifolds”?

- Is there a discrete De Rham Cohomology?

(& why it is important)

- Cohomology of Finite Cyclic Groups (Homological Algebra)

- Duality: Cohomology of Groups and Convolution algebra

- Convolution algebra and FTC for POSets

- Rephrase FTC as De Rham Cohomology

- Conclusions and further developments

Main Idea (connect 3 related frameworks)

- Finite differences and sums on finite abelian groups are just De Rham Cohomology Theory of Differential Forms on discrete manifolds (graphs), e.g. cyclic groups for simplicity.

- The connection with POSets, Mobius inversion (convolution algebras) and Fundamental Theorem of Calculus is well known [4] (and the elementary “tip of the iceberg”).

- For cohomology of finite abelian groups, see [1], [2], [3]

- There are many “machineries’ computing homology and cohomology: Hochschild cohomology Tate cohomology, and via classifying spaces (general formalism: derived functors).

The Homology and Cohomology of Z/nZ

with Integer Coefficients

- The homology of a finite cyclic group A=Z/nZ is:

Hp(Z/nZ;Z)= Z for p=0 (connected)

Z for p=1,3,5, … (odd)

0 for p=2,4,6, … (even)

- It is iso to “algebraist topologist” resolution, using classifying spaces; a complex which can be used is (see [1] for details):

--0--> Z --f--> Z --0-->Z---> … --f-->Z--0-->Z, f(x)=nx

- Cohomology of finite cyclic groups A=Z/nZ is (exchange even/odd): H*(Z/nZ;Z)= Z, 0, Z, 0, Z … (for p=0,1,2,3,...)

An alternative complex for H*(Z/nZ;Z)

- Alternatively, for homology consider the following projective resolution (from presentation Z->Z->Z/nZ; see [3], Th.3.5, p.18), where G=Z/nZ=<x> with generator x:

ZG= group ring, complex (ZG, dk),

d1(x)=x-1 is D(f)=f*𝝁 (POSet framework) <-> Finite difference (fn),

d2(1)=n is I(f)=f*1 (POSet) <-> Finite sums (of values of functions).

Apply HomZG( . , Z) to get the (periodic) cohomology of G.

- Now dualizing d1 & d2 as the differential and integral on functions on Z/nZ (convolution algebra), yields the connection with POSet interpretation (Mobius function); interpreted as FT of Calculus, would be the beginning of De Rham Theory ...

- Recall that for a 1D ”space” M (curve), the higher d-forms are trivial:

0-> F(M;R)->Ω1(M;R) ->0,

H0(M)=R^# components,

H1(M;R)=Closed forms / exact forms.

- Ex. M=S1 (the circle), H0=R=H0 (connected: one component)

H1=Z=H1 (winding number of the path).

Discrete Manifolds” are just … graphs!

- Example: a set S with 6 elements and all possible paths on it (complete graph K6);

- What is a coordinate system (a la Descartes)? => Z6 (paths are periodic) = Z2 x Z3 (Chinese Rem. Th.); => it is 2D

- Graphs are more general models for point-2-point interactions [... then add flow of E/p/info etc.].

- Why is this important? It’s the 1000 miles journey to Finite Strings Theory [Jacobi sums are too similar to Veneziano amplitudes of strings, not to be amplitudes of interacting cycles in a discrete “manifold”].

Back to Math: A=Z/nZ & FTC (first; then De Rham!)

- Finite differences and sums for functions on A:

D:C0(A;Z) ----> C1(A;Z), Int:C1(A;Z) ---->C0(A;Z)

(DF)(x)=F’(x)=F(x+1)-F(x), Int [a,b) f(x)dx = f(a)+...+f(b-1)

- F.T.C. 1) Int [a,b] F’(x)dx=F(b)-F(a);

2) F(x)=Int [a,x) f(t)dt is an A.D. of f(x).

Proof: ...

- How to generalize functions to differentials on graphs G … add measures to functions: f(x)g(x)dx ? [think about applications to Fourier coefficients, Gauss sums …];

- For graphs dim(TpG)=|outgoing edges at p] varies with p ...

Discrete De Rham Cohomology for A=Z/nZ

- Functions F:A->R; differential 1-forms w=f(x)dx

[dx is dual to the “vector” x->x+1, i.e. the edge from x to x+1, when viewing A as a graph with nodes C0 & edges C1]

- Closed forms: Ker d0 = constants; ker d1=C1 (all 1-forms);

- Exact forms: Im(D) = { w s.t. f(x)=F(x+1)-F(x) }

- De Rham cohomology groups H*dR(A;k)= Ker/Im ...

Notes:

1) Z/nZ can be viewed as a D-dim space (D-dimensional torus via CRTh.; generators can be defined as the basis of tangent vectors; or characters as cotangent space and then define derivations etc);

2) See Peter Saveliev - Discrete Calculus (GS project).

Convolution Algebra of Functions

- The dual to a group algebra is the convolution algebra of functions (see Fourier duality).

- Here …

- Claim: The dual of finite difference D and finite sums Int are the convolution operators Df=f*mu, Int(f)=f*1 [special case of the formalism for POSets in the context of Mobius inversion, for example].

The Duality and De Rham’s Theorem

- Stokes Theorem yields an isomorphism between the De Rham cohomology of forms and singular homology of chains

- Finite differences D & Int on functions are dual to group ring construction with “algebraic topologist” complex (multiplication by the generator <-> convolution with Mobus function <-> D)

- The correspondence as a discrete De Rham Theorem

Bibliography

1) GroupProps Wiki - Group cohomology of finite cyclic groups

2) MathExchange - Proof sketch - Cohomology of finite cyclic groups

3) The hard way - An Introduction to Cohomology of Groups by Peter Webb

4) Discrete Fundamental Theorem of Calculus by Nat Thiem (from Math 108 Combinatorics)

5) Discrete differential forms, Disc. Diff. Forms Justin Solomon, …

6) The Theorems of Green-Stokes, Gauss-Bonnet and Poincare-Hopf in Graph Theory, by Oliver Knill

7) see Peter Saveliev: Discrete Calculus, Topology, Intuitive Topology

8) On mathematical thinking by Keith Devlin

De Rham Cohomology of Finite Abelian Groups - Google Slides