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)
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Morse theory is the method of studying the topology of a smooth manifold $M$ by the study of Morse functions $M\to\mathbb{R}$ and their associated gradient flows.
Classical Morse theory centered around simple statements like Morse inequalities, concerning just the Betti numbers. It is useful not only for studying manifolds, but also for studying infinite CW-type spaces homotopically filtered in manifolds, as by Milnor and Bott (especially the stable homotopy of the classical groups) for spaces of paths in a smooth manifold.
Novikov–Morse theory is a variant using multivalued functions. There is also a discrete Morse theory for combinatorial cell complexes.
There are some infinite-dimensional generalizations like Floer instanton homology for 3-dimensional manifolds and also the Hamiltonian variant of Floer homology (and cohomology) for (finite dimensional) symplectic manifolds. An especially well-studied case is that of the cotangent bundle of a differentiable manifold with its standard symplectic structure; this is sometimes called Floer–Oh homology. Floer homology has been partly motivated by Arnold’s conjecture on periodic trajectories in classical mechanics. The symplectic variant of Floer cohomology is related to quantum cohomology.
The founders of Morse theory were Marston Morse, Raoul Bott and Albert Schwarz.
On a smooth manifold $M$, a smooth function $\varphi: M \to \mathbb{R}$ is said to be Morse (or a Morse function) if
The Morse functions on $M$ are dense in most reasonable topologies you could put on $C^{\infty}(M)$. A further condition which is useful, in case $M$ is not compact, is
Together with a (smooth) Riemann structure $g =\langle \cdot,\cdot\rangle$, any real function $\varphi$ on $M$ defines a flow on $M$ by the equation
The Morse functions are notable in that the flows they define have isolated fixed-points with trivially linearizable dynamics, and …[fixme: less vague?]… no other stable cyles.
When $\varphi$ is Morse and coercive, the unstable manifolds of the fixpoints can be arranged into a CW complex $C_{unstable} (\varphi,g)$, canonically homeomorphic to $M$. When $M$ is compact, $\varphi$ and $-\varphi$ are automatically both coercive, and $-\varphi$ induces a dual CW complexe $C_{stable} (\varphi,g)$. Concretely, ….[details].
Let $X \to Y$ be a surjective submersion of compact smooth manifolds, and assume $Y$ is connected. By suitable implicit function theorems, the preimage of any parametrized non-stationary curve $\gamma :(0,1)\to Y$ is a submanifold of $X$, and furthermore the parameter is a Morse function on this submanifold, having no critical points. (It is not coercive). By a little more analysis, the Morse gradient flow is therefore a smooth family of homotopy equivalences. A trivial adjustment of the Riemann structure further allows that the Morse flow sends fibers to fibers diffeomorphically, so that in fact the fibers over neighboring points of $Y$ are diffeomorphic. But since $Y$ is connected, this implies that all the fibers are diffeomorphic, so that $X\to Y$ is a smooth fiber bundle over $Y$.
The restriction to the unit sphere in $\mathbb{R}^{n+1}$ of a generic quadratic form is Morse with $2(n+1)$ critical points — two of each index; and furthermore this restriction clearly descends to $\mathbb{RP}^n$ as a Morse function with $n+1$ critical points, one of each index. It can be shown that this is indeed the minimal collection of critical points supported by real projective space.
Raoul Bott, Morse theory indomitable, Publications Mathématiques de L’IHÉS, 1988, Volume 68, Number 1, Pages 99-114.
Raoul Bott, Lectures on Morse Theory, Old and New, Bull. Amer. Math. Soc. 7 (1982), 331-358.
Raoul Bott, The stable homotopy of the classical groups.
Ann. of Math. (2) 70 1959 313–337.
Daniel Freed, Commentary on “Lectures on Morse Theory, Old and New”, Bull. Amer. Math. Soc., 48(4), October 2011, 517–523
Marco Gualtieri, Course page, lecture notes and links.
Martin Guest, Morse theory in the 1990s, arXiv:math/0104155.
M. M. Postnikov, Введение в теорию Морса — М.: Наука, 1971
Press, 1965.
There is also a variant due to Barannikov, and in a more abstract form due to Viterbo:
S. Barannikov, The framed Morse complex and its invariants, Advances in Soviet Math. 21 (1994), 93-115.
François Laudenbach, On an article by S. A. Barannikov, arxiv/1509.03490
Dorian Le Peutrec, Francis Nier, Claude Viterbo, Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex, Annales Henri Poincaré 14 (2013), 567–610.
The relation to supersymmetric quantum mechanics is due to
Reviews include
Last revised on May 26, 2022 at 15:08:31. See the history of this page for a list of all contributions to it.