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Can anyone familiar with this theorem and its proof let me know how much algebraic topology is involved, and where specifically? I am familiar with a lot of differential geometry, but not many of the algebraic aspects like cohomology, Lie groups, etc. From the ground up could you make it through sphere theorem without any of these tools?

Assuming algebra is required somewhere along the way, would it be more in the way of basic machinery, or in more sophisticated, deep results?

Thanks a lot!

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I do not think the sphere theorem requires much of algebraic topology. You may check De Carmo's book and Tao's blogpost(…) for relevant details. – Kerry Jul 23 '12 at 22:21
It's been a while since I studied the proof. One of the key ideas is that the bound on curvature gives you a bound on the injectivity radius. If I recall correctly, this injectivity radius bound is fairly easy in even dimensions, but requires a Morse theoretic argument in odd dimensions. – Jason DeVito Jul 23 '12 at 23:06

The Sphere Theorem can be presented in different ways. For example Do Carmo doesn't use the Toponogovs theorem at all to describe and prove the Sphere theorem but it requires the basic knowledge of Algebraic Topology like homeomorphisms, homotopies and all these things that appear in Differential Geometry, mostly to help you understand the ideas behind the theorem and its proof (the Cut Locus, injectivity radius etc).

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