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In mathematics, a **microbundle** is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964.^{[1]} 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.

A (topological) ** -microbundle** over a topological space

- the composition
- for every

In analogy with vector bundles, the integer * * is also called the

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.

- Any vector bundle of rank has an obvious
**underlying -microbundle**, where is the zero section. - Given any topological space
**standard trivial microbundle**of rank . Equivalently, it is the underlying microbundle of the trivial vector bundle of rank . - Given a topological manifold of dimension , the cartesian product together with the projection on the first component and the diagonal map defines an
**tangent microbundle**of . - Given an
**pullback (or induced) microbundle**by , together with the projection - Given an
**restricted microbundle**, also denoted by , is the pullback microbundle with respect to the inclusion .

Two -microbundles and over the same space are **isomorphic** (or equivalent) if there exist a neighborhood of and a neighborhood of , together with a homeomorphism commuting with the projections and the zero sections.

More generally, a **morphism** between microbundles consists of a germ of continuous maps between neighbourhoods of the zero sections as above.

An -microbundle is called **trivial** if it is isomorphic to the standard trivial microbundle of rank . The local triviality condition in the definition of microbundle can therefore be restated as follows: for every * * there is a neighbourhood * * such that the restriction is trivial.

Analogously to parallelisable smooth manifolds, a topological manifold is called **topologically parallelisable** if its tangent microbundle is trivial.

A theorem of James Kister and Barry Mazur states that there is a neighborhood of the zero section which is actually a fiber bundle with fiber and structure group , the group of homeomorphisms of fixing the origin. This neighborhood is unique up to isotopy. Thus every microbundle can be refined to an actual fiber bundle in an essentially unique way.^{[2]}

Taking the fiber bundle contained in the tangent microbundle gives the **topological tangent bundle**. Intuitively, this bundle is obtained by taking a system of small charts for * *, letting each chart * * have a fiber * * over each point in the chart, and gluing these trivial bundles together by overlapping the fibers according to the transition maps.

Microbundle theory is an integral part of the work of Robion Kirby and Laurent C. Siebenmann on smooth structures and PL structures on higher dimensional manifolds.^{[3]}

**^**Milnor, John Willard (1964). "Microbundles. I".*Topology*.**3**: 53–80. doi:10.1016/0040-9383(64)90005-9. MR 0161346.**^**Kister, James M. (1964). "Microbundles are fibre bundles".*Annals of Mathematics*.**80**(1): 190–199. doi:10.2307/1970498. JSTOR 1970498. MR 0180986.**^**Kirby, Robion C.; Siebenmann, Laurent C. (1977).*Foundational essays on topological manifolds, smoothings, and triangulations*(PDF). Annals of Mathematics Studies. Vol. 88. Princeton, N.J.: Princeton University Press. ISBN 0-691-08191-3. MR 0645390.

- Gauld, David; Greenwood, Sina (2000). "Microbundles, manifolds and metrisability".
*Proceedings of the American Mathematical Society*.**128**(9): 2801–2808. doi:10.1090/s0002-9939-00-05343-0. MR 1664358. - Switzer, Robert M. (2002).
*Algebraic topology—homotopy and homology*. Classics in Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-42750-6. MR 1886843. See Chapter 14.

- Microbundle at the Manifold Atlas.