What are Betti numbers?
We can now wrap the introductory discussion on algebraic topology and build all the key results needed before we will open a new front this time on differential topology. Some mathematical fireworks will follow when we will marry algebraic and differential topology in the beautiful theory of de Rham cohmology.
We will continue the discussion from last time about holes and introduce three essential concepts: chains, cycles, and boundaries.
Recall the idea of a simplex. Can we add or subtract two simplexes of order p? The answer is yes and this can be visualized as gluing the simplexes together to form a larger geometrical shape.
Now this can be generalized and we define a p-chain as a formal sum of p-simplexes. This is important because the set of p-chains form a group: Cp
In Cp we can define two subgroups:
- Zp whose elements are p-chains with boundary equal with zero.
- Bp whose elements are p-chains which are the boundary of (p+1)-chains.
The groups Zp and Bp may be infinite dimensional, but their quotient group Hp = Zp /Bp is in many cases finite dimensional and its group dimension is called the Betti number.
Zp defines a cycle and Bp defines a boundary. So what does exactly Hp corresponding to? Recall the properties of a hole:
- It has a boundary
- Is not the boundary of anything else
Q: how many holes does a torus have?
(1) shows that a hole corresponds to an element of Zp because one can walk in a closed loop around the hole and return to the starting point. From (2) we see that a hole cannot be an element of Bp because a hole is not a boundary of a filled region. Hence a hole corresponds to an element of Zp but not Bp meaning it belongs to the quotient group Hp (and the groups are named appropriately: C for chains, Z for zero, B for boundary, H for hole, or Homology).
We also see that the boundary operator ∂ is a map between Cp+1 and Cp In fact this map can be extended indefinitely for a particular space (with the understanding that at some point the groups may become trivial):
∂ ∂ ∂
…→ Cp+2 → Cp+1 → Cp → Cp-1 →
When the image of a map becomes the kernel of the next map such a construction is called an exact sequence. Because ∂∂ = 0 (the boundary of a boundary is zero) in homology we have an exact sequence and therefore Hp gives rise to the “Ker modulo Image” pattern.
A similar pattern will be encountered in differential topology because a double differential is zero as well: dd=0. However in there the arrows will be pointing the other way and instead of homology we will have co-homology. The Betty numbers will be the same because the holes of a space do not change if we discuss chains or integration on the space. Roughly speaking cohomology corresponds to integration on a space, and homolog corresponds to the frontier of that integration.