Topological Surgery Theory

     

     In my previous blogs "TOPOLOGY - KING OF MATHEMATICS" & "Topological Space" we came across Topology's beauty and discussed its properties. Now we flash on "Topological Surgery Theory" - a technique used to do surgery (cut - operation - join) on manifolds. 

   To remind you in simple words manifold refers to any object, formally saying" A manifold of dimension n is a Topological space M which is locally homeomorphic to n-dimensional Euclidean space, Hausdroff and has a countable basis.

In topology, surgery theory is a procedure to transform one manifold into another manifold in a controlled way. There are different kinds of surgery on a manifold. 

The formal way is to identify an embedded structure X on the manifold M that we want to cut, then the embedding

φ : X → M

To exclude it, do the cutting operation

M \ (int(φ(X)))

where int is the interior of a set.

For gluing back (if, ∂X = ∂N) use the operation

M ∪_φ|∂N N

Condition, dim X = dim M = dim N = dim (∂N + 1).

Let's look at some examples,

 M = S¹ (circle), X= D¹(line)

        So, φ : D¹ → S¹ is a line on the circle.

Observation : D¹ is a subspace of S¹.

Now do the cutting operation S¹ \ (int(φ(D¹)))

At the end we will get our required one,

You can try yours other.