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I would like to know how to tackle questions of the following type:

Show that $\mathbb{CP}^{2n}$ is not the boundary of any manifold.

Another such question would be:

Let $\iota: S^1 \to S^3$ be a smooth embedding and $K$ be the image of $\iota$. Show that there is a compact orientable surface $F$ embedded in $S^3$ with $K$ as boundary.

More generally, I would like to know what algebraic topological (homology/cohomology/homotopy theory) tools exist that say anything about what spaces can exist as boundaries.

A reference to an appropriate section in, say Hatcher or Bredon that deals with similar questions would be completely fine too.

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    $\begingroup$ Your first question (and more general question) is related to the Stiefel-Whitney number of the manifold being zero. Your second question is essentially asking for the existence of a Seifert surface which Seifert's algorithm gives you always. $\endgroup$ – Dan Rust Aug 1 '18 at 13:54
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    $\begingroup$ To add an emphasis to the comment of @DanRust, the resolutions to your two questions live in pretty independent branches of topology: the resolution of the first is part of algebraic topology; and the resolution of the second is part of geometric topology. This is not too surprising: to prove something doesn't exist one often uses an invariant, and algebraic topology is good at producing invariants; to prove that something does exist you need to construct it, and geometric topology has lots of constructions. $\endgroup$ – Lee Mosher Aug 1 '18 at 14:15
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    $\begingroup$ You might find my answer here useful. $\endgroup$ – Michael Albanese Aug 1 '18 at 15:11
  • $\begingroup$ @DanRust it might be worth noting that the Seifert surface can always be constructed by writing down a map $S^3 \setminus K \to S^1$ which restricts near $K$ to the 'meridian angle' function and taking the preimage of any regular value. This is the 'relative' version of the fact that every codim 1 homology class in a closed oriented manifold may be represented by a smooth submanifold. (Example by way of warning: $H_2(\Bbb{RP}^3;\Bbb Z) = 0$; you can never represent the cycle $\Bbb{RP}^2$, which represents a class in mod-2 homology, by an oriented submanifold.) $\endgroup$ – user98602 Aug 1 '18 at 17:28

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