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By C.T. Dodson, P.E. Parker, Phillip E. Parker

This booklet arose from classes taught via the authors, and is designed for either tutorial and reference use in the course of and after a first path in algebraic topology. it's a guide for clients who are looking to calculate, yet whose major pursuits are in functions utilizing the present literature, instead of in constructing the speculation. normal parts of functions are differential geometry and theoretical physics. we commence lightly, with various photos to demonstrate the basic rules and buildings in homotopy thought which are wanted in later chapters. We express the way to calculate homotopy teams, homology teams and cohomology jewelry of many of the significant theories, certain homotopy sequences of fibrations, a few very important spectral sequences, and all the obstructions that we will compute from those. Our technique is to combine illustrative examples with these proofs that truly advance transferable calculational aids. We supply broad appendices with notes on historical past fabric, wide tables of information, and an intensive index. viewers: Graduate scholars and pros in arithmetic and physics.

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Proof Let M be a star-shaped cmc surface and suppose after a rigid motion that p0 is the origin of coordinates O = (0, 0, 0). The property of being star-shaped is equivalent to the property that we cannot draw a straight line from O which is tangent to M. This means that the support function based on the point O has constant sign on M. Consider the orientation N that points towards W and denote by x the position vector of M. Then the support function is N, x and because N is the inward orientation, we see that N, x < 0 on M.

12 A round sphere is stable. Proof Assume without loss of generality that the sphere is S2 . Consider the spectrum of the Laplacian operator . The first eigenvalue is λ1 = 0 and the eigenfunctions are the constant functions. The second eigenvalue is λ2 = 2 [CH89]. By the Rayleigh characterization of λ2 , 2 = λ2 = min |∇f |2 dS2 : f ∈ C ∞ S2 , 2 dS2 f 2 S S2 S2 f dS2 = 0 . Hence S2 |∇f |2 dS2 ≥ 2 S2 f 2 dS2 for all differentiable function f with 2 2 2 M f dM = 0. Since |σ | = 2 on S , this proves the stability of S .

Consider f ∈ C ∞ (M) with f = 0 on ∂M and M f dM = 0. Extending f to S2 by 0, we obtain a function f˜ ∈ H 1,2 (S2 ) with S2 f˜ dS2 = 0. As S2 is stable by the preceding proposition, I (f ) = I (f˜) ≥ 0. 2. Since a small spherical cap M is included in a hemisphere S + , λ1 (M) > λ1 (S + ) = 2. Thus if M is a small spherical cap or a hemisphere, the variational characterization of λ1 (M) gives 2 ≤ λ1 (M) ≤ |∇f |2 dM 2 M f dM M for all f ∈ C0∞ (M). As |σ |2 = 2 on M, we conclude I (f ) ≥ 0. For a large spherical cap M, λ1 (M) < λ1 (S + ) = 2.

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