A question on smooth 1-manifolds

Question

Consider two people living on two different smooth 1-manifolds $S$ and $T$ as shown in figure 1. The manifold $S$ is a bump function joining the points $A$ and $B$ and the manifold $T$ is formed by joining two identical bump functions at the points $P$ and $Q$. Let the points marked as $H$ denote the homes of the two guys $X$ and $Y$ living on the manifolds $S$ and $T$ respectively with distances (along the curve) from $H$ and $B$ (and $H$ and $Q$) being 40 km.

Let $X$ and $Y$ are equipped with cars which do not have a reverse gear. Every time $X$ starts on a drive from his home $H$ it is certain that he reaches back home after a 80Km drive. But same is not the case with $Y$. It is not sure how much long a drive does he need to return back to his home or will he even ever return back to his home.

My question is what kind of a manifolds are $S$ and $T$ and how are they related. Is there any paradox here which needs to be resolved. Life of $Y$ can be made more complicated by joining the points $P$ and $Q$ with some more bump functions each lying on different planes.

Answer 1

The examples you are giving of S and T don't leave a lot of room for debate: S is an open interval (say $]0,1[$) and T is a circle, $S^1$. Already, if you're adding the points A and B to S, you're turning your manifold into something which is called a manifold with boundary (which isn't a manifold in the technical sense of the term).

I'm not sure what you're looking for, but the 'loopiness' of T is captured in the homology/homotopy of the manifold. S has trivial homology, while $H_1(T) = \mathbb{Z}$ and its fundamental group $\pi_1(T) = \mathbb{Z}$ as well. If you add more 'roads' between P and Q, you're indeed making these groups even larger. (An extra road would yield $H_1(T) = \mathbb{Z}^2$, for example.) The fact that S is 'boring' is because it's contractible. Algebraic topology is the domain which tries to capture these phenomena; I suggest you read the articles http://en.wikipedia.org/wiki/Homology_theory and http://en.wikipedia.org/wiki/Fundamental_group. However, in algebraic topology you're usually not very interested in the exact manifold structure, just in the topology of the space you're describing. A branch of mathematics which does care about manifolds is differential geometry, and in that case you have very similar tools to compute the number of loops etc., see http://en.wikipedia.org/wiki/De_Rham_cohomology.