algebraictopologynotes

# algebraictopologynotes - AN INTRODUCTION TO ALGEBRAIC...

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Unformatted text preview: AN INTRODUCTION TO ALGEBRAIC TOPOLOGY Spring Semester 2008 1. Path Homotopy Definition. Suppose f and f are both maps from a space X into a space Y . Then f is said to be homotopic to f , denoted by f ' f , if and only if there is a map F from X × [0 , 1] into Y such that each of the following holds. (1) F ( x, 0) = f ( x ) for all x in X . (2) F ( x, 1) = f ( x ) for all x in X . The map F is called a homotopy from f to f . Definition. A path from x to x 1 in a space X is a map f from [0 , 1] into X such that f (0) = x and f (1) = x 1 . The points x and x 1 are called the initial point of the path f and the final point of the path f respectively. Definition. Suppose f and f are paths in a space X . Then f is said to be path homotopic to f , denoted by f ' p f , if and only if they have the same initial point x , the same final point x 1 , and there is a map F from [0 , 1] × [0 , 1] into X such that each of the following holds. (1) F ( s, 0) = f ( s ) for all s in [0 , 1]. (2) F ( s, 1) = f ( s ) for all s in [0 , 1]. (3) F (0 ,t ) = x for all t in [0 , 1]. (4) F (1 ,t ) = x 1 for all t in [0 , 1]. The map F is called a path homotopy from f to f . Theorem 1.1. Any path is path homotopic to itself. Theorem 1.2. If a path f is path homotopic to a path f , then f is path homotopic to f . Theorem 1.3. Suppose X and Y are topological spaces, X = A ∪ B where A and B are either both open or both closed, and f : A → Y and g : B → Y are continuous, . If f ( x ) = g ( x ) for x in A ∩ B , then h : X → Y is well-defined and continuous where h is given by h ( x ) = f ( x ) , x ∈ A g ( x ) , x ∈ B . Theorem 1.4. If a path f is path homotopic to a path f , and f is path homotopic to a path f 00 , then f is path homotopic to f 00 . Remark. As a consequence of Theorems 1.1, 1.2, and 1.4, ' p is an equivalence relation. Notice that the proofs of those theorems can be simplified to show that ' is an equivalence relation. Notation. If f is a path in a space X , the path-homotopy equivalence class will be denoted by [ f ]. In other words, [ f ] = { g : g is a path in X and g ' p f } . Definition. Suppose f is a path in X from x to x 1 and g is a path from x 1 to x 2 . The composition of f and g , denoted by f * g , is the path h given by h ( s ) = f (2 s ) for s ∈ [0 , 1 2 ] g (2 s- 1) for s ∈ [ 1 2 , 1] . Theorem 1.5. Suppose f is a path in a space X from x to x 1 and f is a path in X from x 1 to x 2 . If g and g are paths such that f ' p g and f ' p g , then f * f ' p g * g . Remark. Suppose f is a path from x to x 1 and f is a path in X from x 1 to x 2 . As a consequence of Theorem 1.5, the composition of any member of [ f ] with any member of [ f ] is a member of [ f * f ]. Hence we adopt the notation [ f ] * [ f ] = [ f * f ]....
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algebraictopologynotes - AN INTRODUCTION TO ALGEBRAIC...

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