What are Betti numbers?

Intuitive Homology

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: C_p

In C_p we can define two subgroups:

• Z_p whose elements are p-chains with boundary equal with zero.
• B_p whose elements are p-chains which are the boundary of (p+1)-chains.

The groups Z_p and B_p may be infinite dimensional, but their quotient group H_p  = Z_p /B_p is in many cases finite dimensional and its group dimension is called the Betti number.

Z_p defines a cycle and B_p defines a boundary. So what does exactly H_p corresponding to? Recall the properties of a hole:

1. It has a boundary
2. Is not the boundary of anything else

Q: how many holes does a torus have?

A: two

 (1) shows that a hole corresponds to an element of Z_p  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 B_p because a hole is not a boundary of a filled region. Hence a hole corresponds to an element of Z_p but not B_p meaning it belongs to the quotient group H_p (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 C_p+1 and C_p In fact this map can be extended indefinitely for a particular space (with the understanding that at some point the groups may become trivial):

                  ∂            ∂        ∂         

…→ C_p+2 →  C_p+1 → C_p → C_p-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 H_p 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.