Manifolds and cell complexes

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• ConceptTopological manifold

  A topological manifold of dimension $n$ is a topological space that looks like the Euclidean space $\R^n$ near each of its points. It is also required to be Hausdorff and second-countable. It is the basic object of geometric topology. Curves, surfaces and the space-times of physics are manifolds, and smooth and piecewise linear manifolds are topological manifolds with extra structure.

• ConceptAtlas

  An atlas on a topological space is a family of charts whose domains cover the space. A chart is a homeomorphism from an open piece of the space onto an open subset of $\R^n$. Through an atlas, questions about the space can be turned into questions about open sets of $\R^n$. Extra structures on manifolds, such as smooth or piecewise linear structures, are defined by special atlases. In such an atlas, the changes of chart belong to a chosen class of maps.

• ConceptManifold with boundary

  A manifold with boundary is a space that looks near each point either like $\R^n$ or like the closed half-space of $\R^n$. The points of the second kind form the boundary, which is itself a manifold of dimension $n - 1$ without boundary. Closed disks, closed intervals and solid balls are the first examples, and cobordism and surgery theory are built on them.

• ConceptManifold with corners

  A smooth manifold with corners is a space modelled near each point on an open subset of the closed orthant $[0, \infty)^n$. Its changes of chart are smooth. It allows boundary points where several boundary faces meet at an angle, like the vertices of a square or a cube. Products of manifolds with boundary are manifolds with corners. This is one reason the notion is needed, for example in cobordism theory and in Stokes' theorem on cubes.

• TheoremInvariance of domain

  Let $n \ge 1$, let $U$ be an open subset of $\R^n$, and let $f \colon U \to \R^n$ be a continuous injective map. Then $f(U)$ is open in $\R^n$, and $f$ is a homeomorphism from $U$ onto $f(U)$.

• ConceptOrientability

  A manifold is orientable when one can choose an orientation at every point, varying continuously over the whole manifold. An orientation is a sense of "clockwise" in dimension $2$, or of "right-handed" in dimension $3$. The sphere and the torus are orientable; the Möbius strip, the Klein bottle and the real projective plane are not. Orientability is needed to integrate differential forms of top degree, to define the fundamental class and to state Poincaré duality with integer coefficients.