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You are not logged in. #1 20080816 07:54:54
Algebraic topologyI will summarize the results I developed and presented another thread in a more coherent and easiertofollow fashion here. A continuous function is callled a homotopy from f to g. We say that f and g are homotopic iff there exists a homotopy from f to g, and we write ; we can also express the fact that H is a homotopy from f to g by writing . The relation “is homotopic to” is an equivalence relation on , the class of all continuous functions from X to Y. (i) For any , the function is a homotopy from f to itself. (ii) For any , if , then is a homotopy from g to f. (ii) For any , if and , define as follows. Then . the equivalence classes under this equivalence relation are called homotopy classes. #2 20080825 02:52:49
Re: Algebraic topology2. RELATIVE HOMOTOPY Like ordinary homotopy, relative homotopy is an equivalence relation on . Indeed, if , H is just an ordinary homotopy. #3 20080915 02:41:32
Re: Algebraic topology3. PATHS If p is a path in X from a to b and q is a path in X from b to c, the product path is the path from a to c defined by #4 20080915 03:11:45
Re: Algebraic topologyI'm amazed, sounds important I see clearly now, the universe have the black dots, Thus I am on my way of inventing this remedy... #5 20080915 03:32:17
Re: Algebraic topologyIn differential geometry, you also define the set of paths to a point to be the tangent space at that point. The paths are of the form: Such that Where M is your manifold. We then use the tangent space to define what a "derivative" means. "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..." #6 20080915 03:46:22
Re: Algebraic topologyThanks Ricky. 