Supplement - Supplement: The Long Exact Homology Sequence...

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Supplement: The Long Exact Homology Sequence and Applications S1. Chain Complexes In the supplement, we will develop some of the building blocks for algebraic topology. As we go along, we will make brief comments [in brackets] indicating the connection between the algebraic machinery and the topological setting, but for best results here, please consult a text or attend lectures on algebraic topology. S1.1 Defnitions and Comments A chain complex (or simply a complex ) C is a family of R -modules C n , n Z , along with R -homomorphisms d n : C n C n 1 called diferentials , satisfying d n d n +1 = 0 for all n . A chain complex with only Fnitely many C n ’s is allowed; it can always be extended with the aid of zero modules and zero maps. [In topology, C n is the abelian group of n - chains , that is, all formal linear combinations with integer coefficients of n -simplices in a topological space X . The map d n is the boundary operator , which assigns to an n -simplex an n 1-chain that represents the oriented boundary of the simplex.] The kernel of d n is written Z n ( C ) or just Z n ; elements of Z n are called cycles in dimension n . The image of d n +1 is written B n ( C ) or just B n ; elements of B n are called boundaries in dimension n . Since the composition of two successive di±erentials is 0, it follows that B n Z n . The quotient Z n /B n is written H n ( C ) or just H n ; it is called the n th homology module (or homology group if the underlying ring R is Z ). [The key idea of algebraic topology is the association of an algebraic object, the col- lection of homology groups H n ( X ), to a topological space X . If two spaces X and Y are homeomorphic, in fact if they merely have the same homotopy type, then H n ( X ) and H n ( Y ) are isomorphic for all n . Thus the homology groups can be used to distin- guish between topological spaces; if the homology groups di±er, the spaces cannot be homeomorphic.] Note that any exact sequence is a complex, since the composition of successive maps is 0. 1
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2 S1.2 Defnition A chain map f : C D from a chain complex C to a chain complex D is a collection of module homomorphisms f n : C n D n , such that for all n , the following diagram is commutative. C n f n / d n ² D n d n C n 1 f n 1 / D n 1 We use the same symbol d n to refer to the diFerentials in C and D . [If f : X Y is a continuous map of topological spaces and σ is a singular n -simplex in X , then f # ( σ )= f σ is a singular n -simplex in Y , and f # extends to a homomorphism of n -chains. If we assemble the f # ’s for n =0 , 1 ,... , the result is a chain map.] S1.3 Proposition A chain map f takes cycles to cycles and boundaries to boundaries. Consequently, the map z n + B n ( C ) f n ( z n )+ B n ( D ) is a well-de±ned homomorphism from H n ( C ) to H n ( D ). It is denoted by H n ( f ).
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Supplement - Supplement: The Long Exact Homology Sequence...

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