Correction. This example: << For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1).

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should have excluded the origin: << For example, R^n - {0} / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1).

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And just to be clear, an affine function in the previous post means (possibly a restriction of) any map f: R^n -> R^n of the form f(x) = Ax + b where b is in R^n, and A: R^n -> R^n is an invertible linear map. --Dan << Speaking of affine, there's a famous conjecture in differential geometry (by Shiing-Shen Chern): Define an *affine manifold* to be one having an atlas all of whose transition functions are affine. For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1). CONJECTURE: The Euler characteristic of any compact affine manifold is equal to 0. This has remained unresolved for over 50 years.

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