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Determine the homotopy equivalence classes of the sentence "I love the algebraic topology".

I want to learn that how we can define a homotopy on the set of letters of a sentence. Please with an example, explain this subject for me. Thank you.

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Are you asking which letters from the alphabet have the same homotopy type? For example, L and V, of course. Here you have a classification by homeomorphism (2.bp.blogspot.com/_cldxKGOzgeM/SwV2xmssJqI/AAAAAAAACX0/…) and here you have a classification by homotopy type (2.bp.blogspot.com/_cldxKGOzgeM/SwV3lXQVhfI/AAAAAAAACX8/…) –  Sigur Jan 19 '13 at 16:45
    
@Sigur I have no information on alphabet. Please with an example, explain for me –  aliakbar Jan 19 '13 at 16:47
    
I'm afraid your question isn't clear at all. A homotopy (see here) is a certain sort of function, which can intuitively be thought of as a transformation from one function to another. Perhaps you are asking about homotopy equivalence (see here), which is an equivalence relation on topological spaces defined using homotopies? –  Zev Chonoles Jan 19 '13 at 16:47
    
@Sigur Please recall that my question: Determine the homotopy equivalence classes of the sentence "I love the algebraic topology". –  aliakbar Jan 19 '13 at 16:51
    
@ZevChonoles I Please see the above question –  aliakbar Jan 19 '13 at 16:52
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2 Answers

up vote 5 down vote accepted

Two topological spaces $X$ and $Y$ are homotopy equivalent when there are functions $f:X\to Y$ and $g:Y\to X$ such that $g\circ f$ is homotopic to $\mathrm{id}_X$ (the identity function $\mathrm{id}_X:X\to X$) and $f\circ g$ is homotopic to $\mathrm{id}_Y$.

We can treat the letters of the alphabet as topological spaces. For example, the letter O can clearly be thought of as a circle. Similarly, the letter L is homeomorphic to an interval. The other letters can be identified with various combinations of circles and lines; for example, B is like a circle joined to another circle at a point, since it has two "holes", and A is like a circle with two lines attached to it.

A "homotopy type" is just an equivalence class of topological spaces, where the equivalence relation is homotopy equivalence. In other words, two topological spaces are of the same homotopy type when they are homotopy equivalent. We might choose a simple representative of the equivalence class to refer to. So, we would often say

The annulus has the homotopy type of the circle.

even though the sentence

The circle has the homotopy type of the annulus.

is just as valid.

The question might be asking you to

Take the letters occurring in the sentence, and identify which of them belong to which homotopy type

or it might be asking

Take the entire sentence as a single topological space (i.e. as the disjoint union of its constituent letters), and identify the homotopy type of that space.

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+1 for the nice text. –  Sigur Jan 19 '13 at 17:16
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Here you have the alphabet according to homotopy types:

enter image description here

Here you have the alphabet according to homeomorphism:

enter image description here

Take your sentence I love the algebraic topology and identify the classes.

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Excuse me, I'm confused. –  aliakbar Jan 19 '13 at 16:55
    
Please introduce a good book in this area –  aliakbar Jan 19 '13 at 16:56
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@aliakbar, please, tell us: what is an homotopy equivalence class of a sentence? Try to read math.cornell.edu/~hatcher/AT/ATpage.html –  Sigur Jan 19 '13 at 16:58
    
Which page of hatcher? –  aliakbar Jan 19 '13 at 17:03
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