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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.

figure 1

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strange, i can't see the figure and the mathjax doesn't seem to be working for me on this post –  lurscher May 20 '11 at 11:43
    
@Qmechanic : If there are flaws in the question, point out by commenting. You have changed the question too much, the point here is not the content, but how much one can change...a bad question is worth closing or deleting rather than changing it too much as one would wish. –  Rajesh D May 20 '11 at 16:15
    
Dear @Rajesh D. My apologies. Please be sure that I only tried to help with the best of intentions. I must admit that I often find it difficult to say what you are really asking in your questions. –  Qmechanic May 20 '11 at 17:10

1 Answer 1

up vote 1 down vote accepted

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.

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There is a specific reason for choosing bump functions, which i am not able to express. –  Rajesh D May 20 '11 at 10:41
    
Thank you for the answer and valuable suggestions. –  Rajesh D May 20 '11 at 10:41
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@Rajesh: to make it clear: the bump functions don't really matter; formally speaking, the homology functor cannot distinguish between a continuous and a smooth path (bump function). Otherwise, if you have any more questions, feel free to post them. A final word of advice: if you wait a while before accepting an answer, you might get more responses and views, coming from people who might have a different way of looking at your question. –  Gerben May 20 '11 at 12:37

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