microbundle

A (topological) $n$-microbundle over a topological space $B$ (the "base space") consists of a triple $(E,\; i,\; p)$, where $E$ is a topological space (the "total space"), $i:\; B\; \backslash to\; E$ and $p:\; E\; \backslash to\; B$ are continuous maps (respectively, the "zero section" and the "projection map") such that:

1. the composition $p\; \backslash circ\; i$ is the identity of $B$;
2. for every $b\; \backslash in\; B$, there are a neighborhood $U\; \backslash subseteq\; B$ of $b$ and a neighbourhood $V\; \backslash subseteq\; E$ of $i(b)$ such that $i(U)\; \backslash subseteq\; V$, $p(V)\; \backslash subseteq\; U$, $V$ is homeomorphic to $U\backslash times\; \backslash R^n$ and the maps $p\_\{\backslash mid\; V\}:\; V\; \backslash to\; U$ and $i\_\{\backslash mid\; U\}:\; U\; \backslash to\; V$ commute with $\backslash mathrm\{pr\}\_1:\; U\; \backslash times\; \backslash mathbb\{R\}^n\; \backslash to\; U$ and $U\; \backslash to\; U\; \backslash times\; \backslash mathbb\{R\}^n,\; x\; \backslash mapsto\; (x,0)$.

In analogy with vector bundles, the integer $n\; \backslash geq\; 0$ is also called the rank or the fibre dimension of the microbundle. Similarly, note that the first condition suggests $i$ 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 $E$ 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.

• Any vector bundle $p:\; E\; \backslash to\; B$ of rank $n$ has an obvious underlying $n$-microbundle, where $i$ is the zero section.
• Given any topological space $B$, the cartesian product $B\; \backslash times\; \backslash mathbb\{R\}^n$ (together with the projection on $B$ and the map $x\; \backslash mapsto\; (x,0)$) defines an $n$-microbundle, called the standard trivial microbundle of rank $n$. Equivalently, it is the underlying microbundle of the trivial vector bundle of rank $n$.
• Given a topological manifold of dimension $n$, the cartesian product $M\backslash times\; M$ together with the projection on the first component and the diagonal map $\backslash Delta:\; M\backslash to\; M\backslash times\; M$ defines an $n$-microbundle, called the tangent microbundle of $M$.
• Given an $n$-microbundle $(E,\; i,\; p)$ over $B$ and a continuous map $f:\; A\; \backslash to\; B$, the space $f^*E\; :=\; \backslash \{\; (a,e)\; \backslash in\; A\; \backslash times\; E\; \backslash mid\; f(a)=p(e)\; \backslash \}$ defines an $n$-microbundle over $A$, called the pullback (or induced) microbundle by $f$, together with the projection $p:=\; \backslash mathrm\{pr\}\_1:\; f^*E\; \backslash to\; A$ and the zero section $i:\; A\; \backslash to\; f^*E,\; x\; \backslash mapsto\; (x,\; (i\; \backslash circ\; f)(x))$. If $p:\; E\; \backslash to\; B$ is a vector bundle, the pullback microbundle of its underlying microbundle is precisely the underlying microbundle of the standard pullback bundle.
• Given an $n$-microbundle $(E,\; i,\; p)$ over $B$ and a subspace $A\; \backslash subseteq\; B$, the restricted microbundle, also denoted by $E\_\{\backslash mid\; A\}\; =\; p^\{-1\}(A)$, is the pullback microbundle with respect to the inclusion $A\; \backslash hookrightarrow\; B$.

Two $n$-microbundles $(E\_1,\; i\_1,\; p\_1)$ and $(E\_2,\; i\_2,\; p\_2)$ over the same space $B$ are isomorphic (or equivalent) if there exist a neighborhood $V\_1\; \backslash subseteq\; E\_1$ of $i\_1(B)$ and a neighborhood $V\_2\; \backslash subseteq\; E\_2$ of $i\_2(B)$, together with a homeomorphism $V\_1\; \backslash cong\; V\_2$ commuting with the projections and the zero sections.

More generally, a morphism between microbundles consists of a germ of continuous maps $V\_1\; \backslash to\; V\_2$ between neighbourhoods of the zero sections as above.

An $n$-microbundle is called trivial if it is isomorphic to the standard trivial microbundle of rank $n$. The local triviality condition in the definition of microbundle can therefore be restated as follows: for every $b\; \backslash in\; B$ there is a neighbourhood $U\; \backslash subseteq\; B$ such that the restriction $E\_\{\backslash mid\; U\}$ 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 $\backslash R^n$ and structure group $\backslash operatorname\{Homeo\}(\backslash R^n,0)$, the group of homeomorphisms of $\backslash R^n$ 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.{{cite journal | last=Kister | first=James M. | title=Microbundles are fibre bundles | journal=Annals of Mathematics | volume=80 | issue=1 | year=1964 | doi=10.2307/1970498 | pages=190–199 | jstor=1970498 | mr=0180986| url=http://projecteuclid.org/euclid.bams/1183525710 }}

Taking the fiber bundle contained in the tangent microbundle $(M\backslash times\; M,\; \backslash Delta,\; \backslash mathrm\{pr\})$ gives the topological tangent bundle. Intuitively, this bundle is obtained by taking a system of small charts for $M$, letting each chart $U$ have a fiber $U$ 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.{{cite book|first1= Robion C. |last1=Kirby |author1-link=Robion Kirby|first2= Laurent C.|last2= Siebenmann| author2-link=Laurent C. Siebenmann|title=Foundational essays on topological manifolds, smoothings, and triangulations| url=http://www.maths.ed.ac.uk/~aar/papers/ks.pdf| series=Annals of Mathematics Studies|volume= 88| publisher=Princeton University Press| location=Princeton, N.J.|year= 1977|isbn= 0-691-08191-3|mr=0645390}}

• {{cite journal | last1=Gauld | first1=David | last2=Greenwood | first2=Sina | title=Microbundles, manifolds and metrisability| journal=Proceedings of the American Mathematical Society | volume=128 | issue=9 | year=2000 | doi=10.1090/s0002-9939-00-05343-0 | pages=2801–2808 | mr=1664358| doi-access=free }}
• {{cite book | last1=Switzer | first1=Robert M. | title=Algebraic topology—homotopy and homology | publisher=Springer-Verlag | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-42750-6 |mr=1886843 | year=2002}} See Chapter 14.

• [http://www.map.mpim-bonn.mpg.de/Microbundle Microbundle] at the Manifold Atlas.

Category:Geometric topology

Category:Algebraic topology