manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For $i : X \hookrightarrow Y$ an embedding of manifolds, a tubular neighbourhood of $X$ in $Y$ is
a real vector bundle $E \to X$;
an extension of $i$ to an isomorphism
with an open neighbourhood of $X$ in $Y$.
The derivative of $\hat i$ provides an isomorphism of $E$ with the normal bundle $\nu_{X/Y}$ of $X$ in $Y$.
For instance (DaSilva, theorem 3.1).
Moreover, tubular neighbourhoods are unique up to homotopy in a suitable sense:
For an embedding $i : X \to Y$, write $Tub(i)$ for the topological space whose underlying set is the set of tubular neighbourhoods of $i$ and whose topology is the subspace topology of $Hom(N_i X, Y)$ equipped with the C-infinity topology.
If $X$ and $Y$ are compact manifolds, then $Tub(i)$ is contractible for all embeddings $i : X \to Y$.
This appears as (Godin, prop. 31).
(…) propagating flow (…) (Godin).
key application: Pontrjagin-Thom collapse map
Ana Cannas da Silva, §3 of: Prerequisites from differential geometry [pdf]
Ana Cannas da Silva, §6.2 in: Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer (2008) [doi:10.1007/978-3-540-45330-7, pdf]
The homotopical uniqueness of tubular neighbourhoods is discussed in
For an analogue in homotopical algebraic geometry see
See also
Last revised on November 16, 2023 at 09:43:58. See the history of this page for a list of all contributions to it.