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As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.

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    $\begingroup$ It depends on what do they know. If they know some differential topology, then Milnor's Morse Theory book would be my suggestion. If they do not, then teach them some differential topology instead of handle decomposition. $\endgroup$
    – Misha
    Commented Mar 10, 2014 at 18:53

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http://books.google.com/books/about/Differential_topology.html?id=cPZUAAAAYAAJ This is a book of Wallace, Differential Topology, first steps. It is rather detailed, starts from the very beginning and ends with handle decomposition.

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It sounds like Matsumoto's An Introduction to Morse Theory might be precisely what you are looking for. It gets to the heart of the matter quickly, and explains the main ideas very well. I taught a reading course out of it, and was very happy.

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I would suggest Kosinski's differential topology, who does all the necessary differential topology (manifolds, bundles, transversality) before introducing handles around page 120. It is a very nice book: the author goes quickly but rigorously.

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