differential geometry w notes from teacher_Part_82

differential geometry w notes from teacher_Part_82 -...

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Unformatted text preview: Chapter 7 Homology Theory 7.1 Algebraic Preliminaries 7.1.1 Groups • A group is a set G with a binary operation, · , called the group multiplication, that is, 1. associative, 2. has an identity element, 3. every element has an inverse. • A group is Abelian if the group operation is commutative. • For an Abelian group the group operation is called addition and is denoted by + . The identity element is called zero and denoted by 0. The inverse element of an element g ∈ G is denoted by (- g ). • Let G and E be Abelian groups. A map F : G → E is called a homomor- phism if for any g , g ∈ G , F ( g + G g ) = F ( g ) + E F ( g ) , where + G and + E are the group operations in G and E respectively. • In particular, F (0 G ) = E , and F (- g ) =- F ( g ) . 163 164 CHAPTER 7. HOMOLOGY THEORY • Let F : G → E be a homomorphism of an Abelian group G into an Abelian group E . The set of elements of G mapped to the identity element of E is denoted by Ker F = { g ∈ G | F ( g ) =...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_82 -...

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