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
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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The Poincaré–Hopf theorem says that for any vector field $v \in \Gamma(T X)$ with a finite set of isolated vanishing points $\{x_i\}$ on an orientable compact differential manifold $X$, the sum over the $x_i \in X$ of the degrees of the vector in the vicinity of these points, regarded as cohomotopy classes
and called the Poincaré–Hopf index of $f$ at $x_i$
is given by the Euler characteristic, hence by the value of the Euler class on the tangent bundle:
In particular, the existence of a nowhere vanishing vector field (for which the above sum is empty) implies that the Euler characteristic vanishes.
The converse also holds: If the Euler characteristic of a closed manifold vanishes, then a nowhere vanishing vector field exists (e.g. MO:a/47553/58526).
Also: Given any connected closed manifold, the complement of any positive number of disjoint open balls carries a nowhere vanishing vector field (namely: choose any vector field with isolated zeros on the original manifold and then deform continuously to push all these zeros into inside the open balls). Smoothly scaling this vector field to zero towards the centers of these open balls gives a vector field on the original manifold which may be used in the PH-theorem.
Named after Henri Poincaré and Heinz Hopf.
Textbook accounts:
Raoul Bott, Loring Tu, Chapter 11 of Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)
B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, section 15.2 of Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985 (doi:10.1007/978-1-4612-1100-6)
John Milnor, Chapter 6 of: Topology from the differential viewpoint, Princeton University Press, 1997. (ISBN:9780691048338, pdf)
Gerard Walschap, chapter 6.7 of Metric Structures in Differential Geometry, Graduate Texts in Mathematics, Springer 2004
Review:
Alex Wright, Kael Dixon, The Poincaré–Hopf theorem (pdf)
Ariel Hafftka, Differential topology and the Poincaré–Hopf theorem (pdf)
See also:
Discussion in a broader perspective of K-theory and index theorems:
A comment on the version for complex vector fields is in
Generalization to orbifolds:
Last revised on February 23, 2021 at 12:48:38. See the history of this page for a list of all contributions to it.