The glued space of Mobius band and torus

Question

The gluing is made by identifying the boundary circle of the Mobius band with the circle $S^1\times {x_0}$ inside the torus. I just considered it by the fundamental polygons of the two spaces. And it seems the fundamental polygon of the glued space is similar to the connected sum of the two spaces, i.e. the generators of the glued space are just the union of the generators of the two spaces, and the relations are just the union of the relations of the two spaces. So can I just see the glued space as the connected sum of the two spaces?

Answer 1

No, your glued space is not the connected sum of the Mobius band and the torus.

Connected sum is, roughly speaking, a binary operation on connected manifolds. To form the connected sum of $M$ and $N$: choose a closed disc $D_1 \subset M$ with boundary circle $\partial D_1$; choose a closed disc $D_2 \subset N$ with boundary circle $\partial D_2$; then form the quotient space of $M - \text{int}(D_1)$ and $N - \text{int}(D_2)$ by gluing the circles $\partial D_1$ and $\partial D_2$. The gluing you described does not fit this definition.

Furthermore, your space cannot possibly be a connected sum of manifolds, because the connected sum of two manifolds is also a manifold, but your space is not a manifold. If you look at a point $p$ lying on the circle where the boundary of the mobius band was identified with $S^1 \times x_0$, you will see that $p$ has no neighborhood homeomorphic to an open disc in $\mathbb{R}^2$.