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In Milnor's book on Morse theory, he argues on page 14-19 for Theorem 3.2. These pages are not available on the google books preview. On p.16 of this argument, he notes, in his notation, that $e_\lambda \cap M_{c-\epsilon}$ is the boundary of $e_\lambda$ but he must be speaking imprecisely, since even in his picture, this is not true. Besides, being the boundary of something is a relative notion, since it has to do with interiors and closures.

Although this statement is seemingly false, I think that maybe if I could reach the next statement, I could just ignore this one. But the next thing he concludes is that the space $e_\lambda \cup M_{c-\epsilon}$ is homeomorphic to the space obtained by attaching $M_{c-\epsilon}$ to $e_\lambda$ along their intersection via the inclusion map. At least that is what I'm assuming he means. This whole section isn't very clearly spelled out. In any case, I can't see why a claim like that would follow, especially if it has to involve something about boundaries that doesn't seem true.

What seems to be going on here is that Milnor seems to be saying that in general, if $X$ is a topological space, and $A$ and $B$ are subspaces attached in a particular "boundary" way, then $A \cup B$ is homeomorphic to $A$ attached to $B$ in the above way, through the bijective map that takes a member of the union to its equivalence class in the space obtained by attaching.

In the space obtained by attaching the two spaces, the two spaces are "far apart" except for where they're attached. But if we take the union instead and the topology is induced from the manifold, we can have a situation in which one piece affects the topology of the other, say by getting "infinitely close" to it. To make this precise, just consider the topologist's sine curve, $sin(1/x)$ with an extra point at the origin added as $A$, and $B$ will be the line segment from $-(0, -1)$ to $(0, 1)$ in the plane. Here, whatever the boundary is supposed to mean, the only boundary of A should be the origin. But the space obtained via attaching has open sets that intersect the line segment and not the sine curve.

So I am confused as to what Milnor is trying to say here on both claims stated in the first two paragraphs, especially in light of the example provided in the 4th paragraph. Please shed some light on this, thanks!

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I don't have a copy of the text handy, but I think you might be confusing the point set notion of boundary with the boundary of a manifold. The two different ideas of boundaries don't always agree, for example $S^1$ with its usual embedding in $\mathbb{R}^2$ is its own topological boundary, but it doesn't have a boundary as a manifold.

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I considered this, but if you look at his picture, it also isn't fit for the manifold definition. He basically has two two dimensional manifolds linked by a 1-cell, and he says that the 1 cell intersects the 2-manifolds in the "boundary" hence it is attached there. Perhaps by boundary, he means boundary of the one cell in the sense of the endpoints. But then my second problem still stands, if that is the correct interpretation. –  Jeff Jun 19 '12 at 3:59
    
Unfortunately the diagram in question doesn't appear to show up in google books, but if $e_{\lambda}$ is a one cell, I bet that is what he means. He is working with smooth manifolds, right? In that case pathological examples like the topologist's sine curve aren't something that you have to worry about, so your third paragraph is basically true if you change "topological space" to "smooth manifold". Hopefully someone with a copy of the book lying around comes by soon! –  James Cameron Jun 19 '12 at 4:27
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After staring down this proof for a while, I realized what he needs is for the notion of boundary to be the notion of boundary of a cell, as a cell. Thanks for your help. –  Jeff Jun 26 '12 at 1:17
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