Rigged manifold

framed manifold

A smooth manifold with a fixed trivialization of the normal bundle. More accurately, let a smooth $ n $- dimensional manifold $ M $ be imbedded in $ \mathbf R ^ {n+} k $ and let the ( $ k $- dimensional) normal fibration $ \nu $ corresponding to this imbedding be trivial. Any trivialization of the fibration $ \nu $ is called a rigging (framing) of the manifold $ M $ corresponding to this imbedding. Framed manifolds were introduced around 1950 (see [1]) in order to prove that the cobordism groups of $ n $- dimensional framed manifolds lying in $ \mathbf R ^ {n+} k $ are isomorphic to the homotopy groups $ \pi _ {n+} k ( S ^ {n} ) $; the groups $ \pi _ {n+} 1 ( S ^ {n} ) $ and $ \pi _ {n+} 2 ( S ^ {n} ) $ 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 $ T _ {x} M $ of its tangent bundle, with the chosen basis depending differentiably on $ x $.

The frame bundle $ F( M) $ over an $ n $- dimensional smooth manifold $ M $ is the $ n ^ {2} $- dimensional smooth fibre bundle over $ M $( so its total space is of dimension $ n ^ {2} + n $) whose fibre over $ x \in M $ consists of all linear isomorphisms $ T _ {x} M \simeq \mathbf R ^ {n} $. Equivalently, the fibre at $ x $ consists of all ordered bases, also called frames, for $ T _ {x} M $. Thus, more precisely, a framed manifold is a pair $ ( M , s) $ consisting of a smooth manifold $ M $ together with a section $ s : M \rightarrow F ( M) $ 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