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This is a cross post of MSE.

Q1: What does "irreducible manifold" mean (not definition)?

My understanding of "irreducible manifold" is "is not reducible (homotopic or deformation or homeomorph or being prime manifold or something like these) to other manifold". If so, what is the meaning (definition) of reducible manifold?

Q2: What is the origin of this terminology?

Q3: Is there any Book or expository paper about irreducible manifolds in higher dimensions?

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  • $\begingroup$ The basic point is that you can write a non-prime manifold as a nontrivial connected sum; this is the sense in which it can be reduced. For 3-manifolds, prime and irreducible are equivalent except for two examples. Wikipedia says: 'From an algebraist's perspective, prime manifolds should be called "irreducible"''. $\endgroup$ Nov 28 '19 at 19:05
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    $\begingroup$ Regarding high dimensions, generally irreducible manifolds do not exist, this is because the connect-sum operation has some invertible objects -- in dimension 5 and up they are known as homotopy-spheres. $\endgroup$ Nov 28 '19 at 20:17
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    $\begingroup$ These are the theorems of Kervaire and Milnor from the early 60's. Do a Google or library search for "Groups of homotopy spheres" and you should find papers by those two. A significant portion of that work is summarized in Kosinski's "Differential Manifolds" book, as well. $\endgroup$ Nov 28 '19 at 20:54
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    $\begingroup$ What does it mean to ask what a word means, if not its definition? Are you looking for intuition? $\endgroup$
    – LSpice
    Nov 28 '19 at 21:01
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    $\begingroup$ @C.F.G, correct. Symmetric spaces are an extremely structured family of manifolds. In particular (non-trivial) homotopy spheres are not symmetric spaces, I think this is an old result of Wu Chung Hsiang's, from the 60's. $\endgroup$ Apr 21 '21 at 20:14
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Summary of comments and other sources

There are at least 4 similar concepts:

  1. Irreducible smooth manifold: As Ryan Budney said, "Regarding high dimensions, generally irreducible manifolds do not exist, this is because the connect-sum operation has some invertible objects -- in dimension 5 and up they are known as homotopy-spheres." See more in

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

and

Kervaire, Michel A.; Milnor, John W., Groups of homotopy spheres. I, Ann. Math. (2) 77, 504-537 (1963). ZBL0115.40505.

  1. Irreducible symmetric spaces: A symmetric space is said to be irreducible if it is not isometric to a product of positive-dimensional symmetric spaces.

  2. Irreducible vector space: A vector space is said to be irreducible if has no nontrivial invariant subspaces inside it.

and using case 3 we can talk about

  1. Irreducible Riemannian manifolds: Those manifolds where the holonomy has no invariant subspaces.
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    $\begingroup$ Generally speaking "irreducible" as a word in topology is pulled directly from algebra. Say you have a class of topological spaces with a monoidal operation, like oriented manifolds (up to diffeo) and the connect-sum operation, or symmetric spaces and the cartesian product operation. Then irreducible just means objects that can't be written as a product, exactly as in algebra. $\endgroup$ Jul 4 '21 at 21:41

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