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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 Δ h h , h i vol = Z M R i j h i h j vol + ||∇ j h i || 2 ≥ . di ﬀ geom.tex; April 12, 2006; 17:59; p. 193 196 CHAPTER 7. HOMOLOGY THEORY 2. Thus h = 0. ± • Remark . The elements of the ﬁrst 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 ﬀ geom.tex; April 12, 2006; 17:59; p. 194...
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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.
- Spring '10