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I've been asked which of A,B,C...,Z are contractible.

Intuitively, I can see that all but A, B, D, O, P, Q and R are contractible. These are all (except B) homotopy equivalent to a circle, and B homotopy equivalent to 'two circles' (what do I actually mean by this?).

Is this formal enough? How could I make it more formal, if not? How do I show that a circle isn't contractible?

Thanks

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    It's a little tricky to answer this question without knowing what you're allowed to assume (that is, what's already been proved in the class and what hasn't). If I were asking this question on a homework, I probably wouldn't be expecting a formal answer, instead the point is just to test that you intuitively understand the concept.2011-10-13
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    In answer to your question about the letter B, it is homotopy equivalent to the [wedge](http://en.wikipedia.org/wiki/Wedge_sum) of two circles, which we denote by $S^1\vee S^1$. It is not contractible either because $\pi_1(S^1\vee S^1)\cong\mathbb Z*\mathbb Z$, where the $*$ denotes the free product.2011-10-13

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