I wanted to wait until a while after the due date to post the questions from the first homework to avoid potential perception of foul play. It was due last Friday, and I already turned it in, so I think I should be good now. I have my own answers, which I’ll post in comments later, but I want to hear other people’s ideas. These questions mostly pertain to the “Some Theorems about Manifolds” portion of the course.
- Give an example of a topological space that is not homotopy equivalent to any closed manifold. (Hint: think Poincaré duality…)
- Give an example of a smooth real 2-dimensional submanifold of
that it is not homeomorphic to an affine algebraic subvariety of
- Give an example of a closed oriented 4-dimensional manifold that is not homeomorphic to a smooth projective variety. (Hint: think hard Lefschetz…)
Joe Moeller 
I’m guess that S^4, the 4-sphere, should work. I can swear that the S^4 appeared in LaTeX when I first posted my comment! I wonder why it disappeared.
And I think the hard Lefschetz theorem does the job: if S^4 were a complex projective variety, it would imply that the isomorphism between H^0(S^4) and H^4(S^4) was given by multiplying twice by an element of H^2(S^4). But H^2(S^4) = 0 so that’s impossible.
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I’m guessing that S^4, the 4-sphere, should work. I can swear that the S^4 appeared in LaTeX when I first posted my comment! I wonder why it disappeared.
And I think the hard Lefschetz theorem does the job: if S^4 were a complex projective variety, it would imply that the isomorphism between H^0(S^4) and H^4(S^4) was given by multiplying twice by an element of H^2(S^4). But H^2(S^4) = 0 so that’s impossible.
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There’s a harder way to show that S^4 can’t be a complex projective variety: this sphere can’t be made into a complex manifold at all! The only spheres that might be complex manifolds are S^2 (which is) and S^6 (which might be, but this is a famous open question).
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