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Unformatted text preview: 1-forms on M. Proof : 1. Let h be a harmonic 1-form. Then = 1 2 Z M h h , h i vol = Z M R i j h i h j vol + || j h i || 2 . di geom.tex; April 12, 2006; 17:59; p. 193 196 CHAPTER 7. HOMOLOGY THEORY 2. Thus h = 0. Remark . The elements of the rst homology group H 1 ( M , G ) are equiva-lence classes of 1-cycles. The 1-cycles are closed oriented curves (loops) on M . If a closed curve can be deformed to a point, then it is a boundary of a surface (a 2-simplex). That is, a closed curve that can be contracted to a point is a trivial 1-cycle. Corollary 7.5.4 For any simply connected manifold M and any Abelian group G, H 1 ( M , G ) = . di geom.tex; April 12, 2006; 17:59; p. 194...
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- Spring '10