Talk:Clutching construction

Never heard of it

I've never heard this term before. Can someone provide a reliable reference for this usage? Also, I don't really see the need for a whole separate article on this. Almost all of it is already in Alexander trick. --C S (Talk) 23:21, 20 December 2006 (UTC)

I've definitely heard the term before, although usually in connection with constructing vector or principal bundles on the sphere. See Hatcher's (half written) book on Vector Bundles and K-theory for example. -- Fropuff 00:41, 24 December 2006 (UTC)

yeah, but I don't think anybody calls gluing together two balls "clutching construction". Basically it seems all the content should be wiped out and replaced with the valid use for constructing bundles. --C S (Talk) 08:56, 22 January 2007 (UTC)

Agreed. (Sorry, I just came across this page, so I'm adding to the discussion a bit late.) A Google search didn't produce any results that I could find using the term to refer to gluing balls together. It's actually a bit misleading the way it's written on this page because it makes it sound like one might try to glue two seven-balls together along an S^6 to get the exotic spheres, which is not the way Milnor's examples are constructed. I'll put this page on my future (probably far-in-the-future) to-do list. VectorPosse 09:43, 1 March 2007 (UTC)

Hi -- I wrote this page (split off from Alexander trick), but I had confused twisted spheres with the clutching construction (in grad school we'd tended to abuse the term and referred to twisted spheres as a clutching construction). Thanks for pointing this out -- I've fixed it (by completely re-writing it and fixing twisted spheres in the process).

Nbarth 01:52, 2 March 2007 (UTC)

The section on the misuse of the term should be deleted. From what you say (and the lack of sources), it's basically a neologism. --C S (Talk) 14:10, 4 March 2007 (UTC)

This clutching construction or whatever you'd like to call it is basically a half-baked version of the classifying map of a fibre bundle. The reason why you don't see things written up at this level in textbooks is because it's part of a more general theory. Basically, given a fibre bundle F --> E --> B then there is an associated map from the base space B to the classifying space of homeomorphisms of F (denoted B(Homeo(F)) but 'B' means classifying space in this context). The 'clutching construction,' put in a slightly more abstract form essentially *is* the map B ---> B(Homeo(F)). Moreover, the classifying map determines the fibrewise homeomorphism type of the fibre bundle E ---> B. Steenrod's the topology of fibre bundles covers this. I'll take a shot at putting this in the article...later. -rybu

I put some details about my comments above. Maybe my comments should be put in the classifying space page instead, and we should say more explicitly that this clutching construction is the special case of the classifying map in the case the fibre bundle has an atlas with two trivializations...Rybu 10:21, 8 April 2007 (UTC)

One thing I find confusing about this article -- it goes on to say how a clutching construction differs from the constructions of exotic spheres. I don't see how that's the case. But in spirit it's the same thing: you have local trivializations and you construct the global object by gluing together the locally trivial things. Transition function, clutching map, etc, these terms are used in both contexts. Rybu 10:30, 8 April 2007 (UTC)

The unity between these ideas is made very explicit in the Kirby-Siebenmann approach to smoothings of topological and PL manifolds. Smooth structures on a n-manifold M are classified by lifts of the topological tangent bundle 'classifying map' M --> B(Homeo(R^n)) to B(Diff(R^n)). When you combine this with the Cerf-Morlet 'comparison theorem,' it says that the difference between the smooth and topological categories is essentially 'measured' by the homotopy-groups of the group of diffeomorphisms of the compact n-ball, where the diffeomorphisms are required to be the identity on the boundary. Rybu 10:42, 8 April 2007 (UTC)

Rybu: The construction of an exotic sphere Sⁿ you are thinking of is given by a smooth diffeomorphism h: S^n-1 → S^n-1 from the boundary S^n-1 of one Dⁿ to the boundary S^n-1 of another Dⁿ. This map is used to glue the two Dⁿ's smoothly to obtain a smooth structure on a topological Sⁿ; this smooth sphere may be called Σⁿ. Up to diffeomorphism, this structure depends only on the isotopy class of the diffeomorphism h.

By contrast, a clutching map is often used on a sphere possessing a fixed smooth structure in order to define a vector bundle having that sphere as base. The idea is that a vector bundle over a Dⁿ (as for any contractible space) is trivial, so its total space can be expressed as Dⁿ x R^k. Since a smooth sphere Σ = Σⁿ can be thought of as two Dⁿ's glued together (as above), we can think of any vector bundle over Σ as two trivial R^k bundles each over a Dⁿ, but with their fibres over the disks' boundaries identified with each other. This can be done by defining some linear map from the fibre over each point of the boundary of one Dⁿ to the fibre over the identical point (in Σ) in the other Dⁿ.

The two Dⁿ's have a common boundary S^n-1, and each Dⁿ has its own R^k fibre above each point of this S^n-1. The clutching map takes each point x of this S^n-1 and maps the R^k fibre over x of one n-disk's trivialization to the R^k fibre over x of the other disk's trivialization. It can be shown that it is sufficient to choose, for each x in S^n-1, a rotation in SO(k) to effect this identification. (Actually, if n=1, we will need maps from S⁰ into O(k) to get the two distinct bundles.)

Thus the clutching map is given by some continuous map f: S^n-1 → SO(k). It turns out that if two such clutching maps f, g are homotopic, then they will give rise to equivalent bundles. (There can also be non-homotopic clutching maps that give rise to equivalent bundles.)Daqu (talk) 19:00, 10 July 2014 (UTC)