# A Torus with two Möbius Strips

1. Dec 16, 2016

### dkotschessaa

Another zany homology question:

Just let me know if I have my labeling scheme right so far.

I have a torus. I have cut two holes into it and am attaching Möbius strips around the holes. (Clearly we are not in 3 dimensions?)

My Torus is represented by a polygon with the labeling scheme $a b a^{-1}b^{-1}$ and my strips are labeled $a b c b$ as follows (I'll subscript them after pasting)

I'll subscript the torus with 1s, and the 2 mobius strips will be subscripted with 2s and 3s.

I think I cut a hole in the Torus just by cutting a corner off of the polygon. I'll attach the strips to attain the following monstrosity:

I end up with a long labeling scheme:

$a_1 b_1 c_1 b_2 b_1 b_3 c_3 b_3 a_1^{-1} b_1^{-1}$

Before I say any further, is this what cutting a hole in a Torus and gluing a Möbius band even would look like? Obviously I can't picture this outside the polygons here. Thanks for your patience.

-Dave K

2. Dec 16, 2016

### zinq

"Before I say any further, is this what cutting a hole in a Torus and gluing a Möbius band even would look like?"

Do you mean "a" Möbius band, or "two" of them?

3. Dec 16, 2016

### dkotschessaa

Two.

4. Dec 19, 2016

### dkotschessaa

Sorry, that's not what I end up with. I get:

$a_1 b_2 c_2 b_2 b_1 b_3 c_3 b_3 a_1^{-1} b_1^{-1}$

Still, if anyone can let me know if I have even this preliminary step right I would appreciate it.

Might as well keep going.

When I relabel I get:

$a b c d a^{-1}b^{-1}$

Which still isn't something I know from the classification theorem.

5. Dec 20, 2016

### lavinia

I don't see how you diagram accomplishes what you want. For instance, once you cut the corner s off the identification of $b1$ no longer makes sense. Can you explain this better?

6. Dec 22, 2016

### dkotschessaa

Sorry, I didn't realize there'd been an update to this thread. Thought I was going to have to go to stackexchange. ::shudder::

I admit that I am confused as to how to properly make this hole with regards to the polygonal region. I will revisit.

-Dave K

7. Dec 22, 2016

### dkotschessaa

Nope. Don't know how to do it. I am thinking that cutting a hole in a torus equates to cutting a corner off the torus, which is really like adding a side. i.e. when Munkres glues two tori together:

So why doesn't my $b_1$ make sense?

8. Dec 22, 2016

### lavinia

It might make sense. I found it confusing because you cut off part of one $b_{1}$ that is identified to the other. So where does the identification go?

9. Dec 22, 2016

### dkotschessaa

Of course in a regular torus, the $b_1$ would go back to the other $b_1$, The $a_1$ similarly. The Mobius strips would be joined up in their usual way, though I don't know how this effects the overall "shape". I certainly can't visualize it! But at least according to the question, this thing should be able to be classified, so it's an n-fold torus, n-fold projective plane or a sphere, although it certainly looks like something much weirder to me.

-Dave K

10. Dec 23, 2016

### lavinia

If you mean the diagram for a torus with a hole in it that you showed here in post 7 then nothing is wrong with $b_1$.

The curve $c$ is a closed loop and you have attached a Mobius band to it using a half the boundary of the band which is not a closed loop. So the band gets folded and is no longer a surface with a boundary. Is that what you want?

One thing that can be done is to attach the entire boundary of the Mobius band to the curve $c$. This lines up two boundary circles so that the resulting glued together space is a closed surface without boundary. Can you redraw this space to make a polygon with edges identified?

I think by Van Kampen's Theorem that the relation will be $aba^{-1}b^{-1}c^{-2}$.

Last edited: Dec 23, 2016