In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situations where they would not ordinarily be thought to exist. For example, the tangent bundle is defined for a smooth manifold but not a topological manifold; use of microbundles allows the definition of a topological tangent bundle.
Definition A (topological) n {\displaystyle n} -microbundle over a topological space B {\displaystyle B} (the "base space") consists of a triple ( E , i , p ) {\displaystyle (E,i,p)} , where E {\displaystyle E} is a topological space (the "total space"), i : B → E {\displaystyle i:B\to E} and p : E → B {\displaystyle p:E\to B} are continuous maps (respectively, the "zero section" and the "projection map") such that:
the composition p ∘ i {\displaystyle p\circ i} is the identity of B {\displaystyle B} ; for every b ∈ B {\displaystyle b\in B} , there are a neighborhood U ⊆ B {\displaystyle U\subseteq B} of b {\displaystyle b} and a neighbourhood V ⊆ E {\displaystyle V\subseteq E} of i ( b ) {\displaystyle i(b)} such that i ( U ) ⊆ V {\displaystyle i(U)\subseteq V} , p ( V ) ⊆ U {\displaystyle p(V)\subseteq U} , V {\displaystyle V} is homeomorphic to U × R n {\displaystyle U\times \mathbb {R} ^{n}} and the maps p ∣ V : V → U {\displaystyle p_{\mid V}:V\to U} and i ∣ U : U → V {\displaystyle i_{\mid U}:U\to V} commute with p r 1 : U × R n → U {\displaystyle \mathrm {pr} _{1}:U\times \mathbb {R} ^{n}\to U} and U → U × R n , x ↦ ( x , 0 ) {\displaystyle U\to U\times \mathbb {R} ^{n},x\mapsto (x,0)} . In analogy with vector bundles, the integer n ≥ 0 {\displaystyle n\geq 0} is also called the rank or the fibre dimension of the microbundle. Similarly, note that the first condition suggests i {\displaystyle i} should be thought of as the zero section of a vector bundle, while the second mimics the local triviality condition on a bundle. An important distinction here is that "local triviality" for microbundles only holds near a neighborhood of the zero section. The space E {\displaystyle E} could look very wild away from that neighborhood. Also, the maps gluing together locally trivial patches of the microbundle may only overlap the fibers. The definition of microbundle can be adapted to other categories more general than the smooth one, such as that of piecewise linear manifolds, by replacing topological spaces and continuous maps by suitable objects and morphisms.
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