nLab
Banach manifold

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Manifolds and cobordisms

Contents

Idea

A notion of infinite-dimensional manifold. A Banach manifold is a manifold modelled on Banach spaces. By default, transition maps are taken to be smooth.

Properties

As absolute neighbourhood retracts

Example

Every paracompact Banach manifold is an absolute neighbourhood retract.

By Palais 1966, Cor. to Thm. 5 on p. 3.

Embedding into the category of diffeological spaces

The category of smooth Banach manifolds has a full and faithful functor into the category of diffeological spaces. In terms of Chen smooth spaces this was observed in (Hain). For more see at Fréchet manifold – Relation to diffeological spaces.

References

For general references see at infinite-dimensional manifold.

Aspects of the homotopy theory of Banach manifolds:

The full subcategory embedding into the category of diffeological spaces:

  • Richard Hain, A characterization of smooth functions defined on a Banach space, Proc. Amer. Math. Soc. 77 (1979), 63-67 (web, pdf)

Last revised on September 19, 2021 at 02:25:40. See the history of this page for a list of all contributions to it.