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I really would like to learn about the foundations of high-dimensional topology, the study of manifolds of dimensions 5 or greater. I learned the algebraic topology, but much of focus in what I learned in topology, such as geometric topology, has been low-dimensional manifolds. I concluded that I am intensely more interested about high-dimensional manifolds than low-dimensional manifolds; unfortunately, it seems that there are no introductory textbooks or monographs about high-dimensional manifolds.

Where should I begin with? What topics should I study that are core of high-dimensional topology? I have a feeling that surgery and cobordism theories are important, but I am not sure. Should I instead trying to read past research articles than specific textbooks?

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    $\begingroup$ Milnor's little book "Lectures on the $h$-cobordism theorem" is, like most Milnor books, quite beautiful. $\endgroup$
    – Lee Mosher
    Jun 17, 2020 at 2:32
  • $\begingroup$ I see! What topics beside cobordism are central to high-dimensional topology, as there seems no book about it $\endgroup$
    – Phoenix13
    Jun 17, 2020 at 2:33
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    $\begingroup$ The key word to look up is "Surgery Theory". Ranicki's "Algebraic and Geometric Surgery" is a pretty accessible introduction. $\endgroup$
    – Connor Malin
    Jun 17, 2020 at 4:15

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The only (I am quite sure) textbook for higher-dimensional manifold topology is

Antoni A. Kosinski, Differential manifolds, Pure and Applied Mathematics, 138. Boston, MA: Academic Press. xvi, 248 p. (1993). ZBL0767.57001.

This edition is expensive, but the book was also republished in paperback form by Dover (2007) which is affordable.

The book even has a fair number of exercises. It starts very gently (at the same level and pace as most textbooks on differential topology) but eventually gets to advanced topics such as the h-cobordism theorem and surgery theory. Of course, as Lee Mosher suggested, you cannot go wrong with Milnor's "h-cobordism" book, but it was not written as a textbook.

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