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I'm trying to prove that $\overline{st_K(x)}$ is a cone on $lk_K(x)$, but can't seem to get anywhere!

I know how to construct a topological cone given a space $X$. However I don't know any way to test whether a space is a cone on another space. Could someone give me a hint for this, and possibly outline some methods I might employ in the future for similar problems?

Many thanks!

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1 Answer 1

up vote 1 down vote accepted

Draw a picture. Alternately, refer to the picture at wikipedia. The easiest way to prove something is a cone based on some space is to figure out exactly what cone it is (in this case, the cone on $lk_K(x)$ from $x$ itself). In order to prove this you'll need to go to the definitions of the objects involved.

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Thanks Paul. I definitely see why it's intuitively true, but I've written down the definition and I can't see how to come up with a conclusive proof. I guess my question really is - if $X=lk(x)$ then find an explicit homeomorphism $CX \rightarrow \overline{st(x)}$. Any ideas what my homeomorphism should be for a general simplicial complex of dimension $n$? Thanks! –  Edward Hughes May 3 '12 at 22:07
@hgbreton It mostly depends on how you defined simplices, but there should be a way to construct a natural isomorphism between any n-simplex and the cone from one of its (n-1)-faces to the vertex not included in that face. –  Paul Z May 4 '12 at 18:14

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