Topology - Gluing two handlebodies by the identity

In summary, the author is having a hard time understanding how to identify the boundaries of two handlebodies of equal genus using the identity mapping. The author thinks that if the two handlebodies share a common boundary, then the identity mapping would map a point to itself and the problem is that this is not the case. The author thinks that the fundamental group of each of the handlebodies is the free group with g generators and this is what is expected from the Seifert-Van-Kampen theorem. However, the author is not sure if this is what is actually happening.
  • #1
barbutzo
8
0
Hello all,

I have a question I'm having a hard time with in an introductory Algebraic Topology course:
Take two handlebodies of equal genus g in S^3 and identify their boundaries by the identity mapping. What is the fundamental group of the resulting space M?

Now, I know you can glue two handlebodies of equal genus to create S^3 itself - but that's gluing is not done by the identity mapping.
Intuitively, it would seem I would get the free group with g generators - it's as if I'm identifying the generators of the fundamental groups of both handlebodies. Thing is, when I'm trying to formalize that notion using the Seifert-Van-Kampen theorem - it doesn't turn out right. If I'm trying to use SVK then after identifying the boundaries, the two handlebodies cover the space, and their intersection is a connected sum of tori. The induced homomorphism from their intersection to each of the handlebodies maps all elements of the intersections fundamental group to e, as they are all equivalent to a point once you can move the loops through the solid handlebody. Now, since the fundamental group of each of the handlebodies is the free group with g generators, it follows from SVK that the fundamental group of M would be the free group with 2g generators (just the free product of both of their fundamental groups, seeing as the intersection didn't create any relations), which, to me, seems twice as much as is correct :)

Any ideas where I'm going wrong?
 
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  • #2
What is meant here by the identity mapping? I would think that the identity mapping would map a point to itself but that would only be a "gluing" if the two handelbodies shared a common boundary.
 
  • #3
The meaning of identity here can be seen this way - suppose we start with one handlebody, and duplicate it to create the second one. Now the identity means mapping each point in the original to the point it was duplicated to in the second.
 
  • #4
OK, I think I've worked out my problem. Is it considered acceptable here to post my final solution to my question?
 
  • #5
Sure, go ahead. I'd be interested in seeing it.
 

FAQ: Topology - Gluing two handlebodies by the identity

1. What is topology?

Topology is a branch of mathematics that deals with the properties and relationships of geometric objects that remain unchanged when they are stretched, twisted, or bent, but not torn. It is often described as the study of the "shape" of space.

2. What is a handlebody in topology?

A handlebody is a three-dimensional shape with a number of "handles" attached to it. In topology, a handle is a higher-dimensional analog of a hole, and a handlebody is a space that is formed by attaching handles to a solid ball. Each handle can be thought of as a cylinder attached to the ball, which is then "glued" back together along the boundary of the cylinder.

3. How do you glue two handlebodies by the identity in topology?

To glue two handlebodies by the identity, we first take two handlebodies and remove a small open disk from each of them. Then, we identify the boundaries of these disks and glue the two handlebodies together along this boundary. This creates a new handlebody that has one more handle than the original two handlebodies.

4. What is the significance of gluing two handlebodies by the identity in topology?

Gluing two handlebodies by the identity is a way to construct higher-dimensional spaces from lower-dimensional ones. This process is important in understanding the topology of more complicated spaces, such as manifolds, and is also used in the study of knot theory and the classification of surfaces.

5. How does gluing two handlebodies by the identity affect the Euler characteristic?

The Euler characteristic is a numerical invariant in topology that is used to classify surfaces. When two handlebodies are glued by the identity, the Euler characteristic of the resulting space is equal to the sum of the Euler characteristics of the two handlebodies. This is because the process of gluing essentially adds a handle to the space, which increases the Euler characteristic by 1.

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