Homology: the skinny

I’d love to get a picture of a triangulated torus in here but I’ve tried for hours and can’t do it.

Homology is a rather esoteric branch of topology concerning holes in shapes (which can have any number of dimensions, not just two or three.  It is very easy to get bogged down in the large number of definitions and algebra without understanding what is really going on.  I certainly did.

 

The following explains what is really going on underneath the massive amounts of algebra (chains, cycles, chain groups, Betti numbers, cohomology, homology groups etc. etc.) required to understand homology.

The doughnut (torus) you see just above is hollow like an inner tube not solid like a donut.  So it is basically a 2 dimensional surface in 3 dimensional space.  Topology ignores what its objects of study (topological spaces) are embedded in, although they all can be embedded in ‘larger’ spaces, just as the 2 dimensional torus can be embedded in good old 3 dimensional space.

 

Homology allows you to look for holes in topological spaces in any dimension.  How would you find the hole in the torus without looking at it as it sits in 3 dimensioal space.

 

Look at the figure.  Its full of intersecting lines.  It is an amazingly difficult to prove theorem that every 2 dimensional surface can be triangulated (e.g. points placed on it so that it is covered with little triangles).  There do exist topological objects which cannot be triangulated (but two dimensional closed surfaces like the torus are not among them).

 

The corners of the triangles are called vertexes.  It’s easy to see how you could start at one vertex, march around using the edges between them and then get back to where you started.  Such a path is called a cycle.  Note that a cycle is one dimensional not two.

 

Every 3 adjacent vertices form a triangle. Paths using the 3 edges between them form a cycle.  This cycle is a boundary (in the mathematical sense) because it separates the torus into two parts.  The cycle is one dimensional because all you need is one number to describe any point on it.

 

So far incredibly trivial?  That’s about all there is to it.

 

No go up to the picture and imagine the red and pink circles as cycles using as many adjacent vertices as needed (the vertices are a bit hard to see). Circle
Neither one is a boundary in the mathematical sense, because they don’t separate the torus into two parts.

 

 

Each one has found a ‘hole’ in the torus, without ever looking at it in 3 dimensions.

 

 

So this particular homology group is the set of cycles in the torus which don’t separate it into two parts.

 

 

Similar reasoning allows you to construct paths made of 3 dimensional objects (say tetrahedrons instead of two dimensional triangles) in a 4 dimensional space of  your choice.  Some of these are cycles separating the 4 dimensional space into separate parts and others are cycles which don’t.  This allows you to look for 3 dimensional holes in 4 dimensional spaces.

 

 

Of course it’s more complicated than this. Homology allows you to look for any of the 1, 2, . . , n-1 dimension holes possible in an n dimensional space — but the idea is the same.

 

There’s tons more lingo to get under your belt, boundary homomorphism, K complex, singular homology, p-simplex, simplicial complex, quotient group, etc. etc. but keep this idea in mind.

 

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Comments

  • Tom  On March 27, 2018 at 3:33 pm

    The image appears after I touch the square in the original post

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