## 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**we can define two subgroups:_{p}**Z**whose elements are p-chains with_{p}**boundary equal with zero**.**B**whose elements are p-chains which are_{p}**the boundary of (p+1)-chains**.

The groups

**Z**and_{p}**B**may be infinite dimensional, but their_{p}**quotient group H**/_{p }= Z_{p}**B**is in many cases finite dimensional and its group dimension is called the_{p }**Betti number.****Z**defines a

_{p}**cycle**and

**B**defines a

_{p}**boundary. So what does exactly H**Recall the properties of a hole:

_{p}corresponding to?- It has
a boundary
- 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**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_{p}**B**because a hole is not a boundary of a filled region. Hence a hole corresponds to an element of_{p }**Z**_{p}**but not B**meaning it belongs to the_{p}**quotient group H**(and the groups are named appropriately: C for chains, Z for zero, B for boundary, H for hole, or_{p}**Homology**).
We also see that the boundary operator ∂ is a map between

**C**and_{p+1}**C**In fact this map can be extended indefinitely for a particular space (with the understanding that at some point the groups may become trivial):_{p}
∂ ∂ ∂

…→

**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**. Because ∂∂ = 0 (the boundary of a boundary is zero) in homology we have an exact sequence and therefore**__exact sequence__**H**gives rise to the_{p}**“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.
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