Manifold Atlas:Definition of “manifold”

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m (moved Definition of “manifold” to MediaWiki:Definition of “manifold”: Part of the structure of the Atlas.)

Revision as of 21:39, 17 September 2009

1 Introduction

This page defines the term “manifold” as used in the Manifold Atlas. We assume that all manifolds are of a fixed dimension n.

Definition 1.1.

An n-dimensional manifold
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is a second countable Hausdorff space for which every point x \in M has a neighbourhood U_x homeomorphic to an open subset of \Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}.
  • The interior of
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    , denoted \mathrm{int}(M), is the subset of points for which U_x \subset \Rr^n.
  • The boundary of
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    , written \partial M, is the complement of the interior of
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    .
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    is called closed if
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    is compact and \partial M is empty.
A manifold
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as above is often called a topological manifold for emphasis or clarity.

Typically, but not necessarly, the word “manifold” will mean "topological manifold with extra structure", be it piecewise-linear, smooth, complex, symplectic, contact, Riemannian, etc. The extra structure will be emphasised or suppressed in notation and vocabulary as is appropriate. We briefly review some common categories of manifolds below.

2 Atlases of charts

We give a unified presentation of piecewise linear, smooth and complex manifolds. Recall that a chart on a topological manifold
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is a homeomporphism \phi_\alpha : U_\alpha \to V_\alpha from an open subset U_\alpha of
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to an open subset V_\alpha of \Rr^n_+. The transition function defined by two charts \phi_\alpha and \phi_\beta is the homeomorphism
\displaystyle  \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).
An atlas for
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is a collection of charts A = \{ (U_\alpha, \phi_\alpha)\} such that the U_\alpha cover
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.

Let \Cat denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate C^\infty maps. That is we require the every \phi_{\alpha, \beta} to be either piecewise linear, smooth of class C^\infty or holomorphic. An atlas is a \Cat Atlas if every transition function defined by the that atlas is a \Cat function. \Cat atlases are compatible if their union again forms a \Cat atlas and by Zorn's Lemma each \Cat atlas defines a unique maximal \Cat atlas.

Definition 2.1. A \Cat-manifold (M, A) is a manifold
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together with a maximal \Cat atlas A. A \Cat-isomorphism (M, A) \cong (N, B) is a homeomorphism f: M \cong N which is a \Cat morphism when viewed in every pair of charts in A and B.
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