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For Differential Geometry there's this wonderful book called "Analysis and Algebra on DIfferentiable Manifolds", which is a big problem book containing problems (and solutions).

For a self-studying student this is particularly helpful because these sorts of books provide us with a good feedback loop. If we attempt a problem, get stuck on and can't solve it, we can always read the first few lines of a solution to see a hint on where to go next, and if we are really stuck we can check the full solution.

Do such books exist for Algebraic Topology, or Differential Topology? Or is the best method to gain a similar sort of feedback loop to read the theorems and examples in a book like Hatcher or Guillemin and Pollack and attempt to prove the theorems and examples by hand?

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    $\begingroup$ "Or is the best method to gain a similar sort of feedback loop to read the theorems and examples in a book like Hatcher or Guillemin and Pollack and attempt to prove the theorems and examples by hand?" or use google or this site. $\endgroup$ – PVAL-inactive Sep 27 '17 at 21:39
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You should check Elementary topology problem textbook, freely available in the English noproof version. An English 2nd ed. with proofs is for sale, but the first ed. (with proofs) is also freely available here (toward the bottom).

Part 2 deals with algebraic topology

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