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**Paths**

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**JaneFairfax****Member**- Registered: 2007-02-23
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Let

be defined by . Then p is a path in from (1,0) to (−1,0). In general:*Last edited by JaneFairfax (2008-07-03 13:39:02)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Let

where q(Note that F[sub]θ[/sub] is a path from (1,0) to (−1,0) for each θ ∊ [0,1] and F[sub]0[/sub](*t*) = p(*t*) and F[sub]1[/sub](*t*) = q(*t*) for each *t* ∊ [0,1]. F[sub]θ[/sub] is a semiellipse with semimajor axis 1 and semiminor axis |1−2θ| (except when θ = ½, when its a straight-line path).

Now, what happens when θ varies continuously from 0 to 1? Yes, the path p transforms continuously to q via the various intermediate paths F[sub]θ[/sub]. We say that the path p is homotopic to q in

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

The function *F* is called a homotopy from *f* to *g*. We can write

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

And we write

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

I see you're starting to get into algebraic topology. Is this self study? What book are you using?

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Yes, its mainly self-study but any help from anybody will be greatly appreciated.

Im using *A First Course in Algebraic Topology* (2005) by B.K. Lahiri (one of the books I bought at Waterstones next to University College London on my visit to Central London the other day). I hate to say that its not very well written (though its a second edition and the author says mistakes in the original edition have been corrected). Hopefully they are not serious mistakes, and a person like me should have little or no problem seeing them for what they are.

Im also supplementing my reading with whatever material I can find online.

*Last edited by JaneFairfax (2008-07-05 08:58:34)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

By the way, MathsIsFun, if you need a page on algebraic topology on the website, I can write one up for you up to and including the defintion of fundamental group.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Let *a*, *b*, *c* be points in a topological space *X*. Suppose

Basically,

is a path fromThen we have this reault:

It must be noted that it is important for the homotopies to be relative to {0,1}. If *p* and *q* (likewise *r* and *s*) are only homotopic, not homotopic relative to {0,1}, then the product paths

Here is an example to show why. Let

and define paths as follows:Then *p* and *q* (likewise *r* and *s*) are only homotopic, not homotopic relative to {0,1} and the product paths

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Let

. The null path atOffline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Let

be a path fromOffline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

*Last edited by JaneFairfax (2008-07-23 21:14:02)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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