Rigged manifold

framed manifold

A smooth manifold with a fixed trivialization of the normal bundle. More accurately, let a smooth -dimensional manifold be imbedded in and let the (-dimensional) normal fibration corresponding to this imbedding be trivial. Any trivialization of the fibration is called a rigging (framing) of the manifold corresponding to this imbedding. Framed manifolds were introduced around 1950 (see [1]) in order to prove that the cobordism groups of -dimensional framed manifolds lying in are isomorphic to the homotopy groups ; the groups and have been computed along these lines.

References

[1]	L.S. Pontryagin, "Smooth manifolds and their application in homotopy theory" Trudy Mat. Inst. Steklov. , 45 (1955) (In Russian)

Comments

The phrase "framed manifold" is also used to denote a differentiable manifold together with a basis in each fibre of its tangent bundle, with the chosen basis depending differentiably on .

The frame bundle over an -dimensional smooth manifold is the -dimensional smooth fibre bundle over (so its total space is of dimension ) whose fibre over consists of all linear isomorphisms . Equivalently, the fibre at consists of all ordered bases, also called frames, for . Thus, more precisely, a framed manifold is a pair consisting of a smooth manifold together with a section of the frame bundle. Such a section is called a framing.

Quite generally, of course, the word "frame" is used as a substitute for basis in a vector space. The terminology derives from the fact that a basis in space-time provides a frame of reference in the sense of mechanics.

References

[a1]	J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965)
[a2]	J.W. Milnor, "A survey of cobordism theory" L'Enseign. Math. , 8 (1962) pp. 16–23
[a3]	R. Thom, "Quelque propriétés globales des variétés différentiables" Comm. Math. Helvet. , 28 (1954) pp. 17–28
[a4]	J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)
[a5]	C.T.J. Dodson, "Categories, bundles, and spacetime topology" , Kluwer (1988) pp. 94ff
[a6]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. 98