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
manifolds and cobordisms
cobordism theory, Introduction
The torus is the manifold (a smooth manifold, hence in particular also a topological manifold) obtained as the quotient
of the Cartesian plane, regarded as an abelian group, by the subgroup of pairs of integers.
As a topological space this is the quotient topological space obtained from the square by identifying opposite sides:
graphics grabbed from Lawson 03
More generally, for $n \in \mathbb{N}$ any natural number, the $n$-torus is
For $n = 1$ this is the circle.
In this fashion each torus canonically carries the structure of an abelian group, in fact of an abelian Lie group. Notice that regarded as a group the torus carries a base point (the neutral element).
According to SGA3, for $X$ a base scheme then a 1-dimensional torus (in the sense of tori-as-groups) over it is a group scheme over $X$ which becomes isomorphic to the multiplicative group over $X$ after a faithfully flat group extension.
In (Lawson-Naumann 12, def. A.1) this is called “a form of” the multiplicative group over $X$.
By (Lawson-Naumann 12, prop. A.4) the moduli stack of 1-dimensional tori $\mathcal{M}_{1dtori}$ in this sense is equivalent to the delooping of the group of order two:
The single nontrival automorphism of any 1-dimensional toris here is that induced by the canonical automorphism of the multiplicative group
which is the inversion involution (given by sending any element to its inverse element).
As a homotopy type the torus is the product of two copies of the circle.
In homotopy type theory the torus can be formalized as the higher inductive type generated by a point base
, two paths, $p$ and $q$, from base
to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown (Sojakova15) that this type is equivalent to the product of two circles. For a treatment in cubical type theory, see (Licata-Brunierie).
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The stable homotopy type of the torus is the wedge sum of 2 circles and one 2-sphere (eg. Freed & Moore 13, Thm. 11.8):
The moduli stack of 1-dimensional tori in algebraic geometry is discussed (as the cusp point inside the moduli stack of elliptic curves) in
Discussion of tori the homotopy type of the torus in homotopy type theory is in
Kristina Sojakova, The equivalence of the torus and the product of two circles in homotopy type theory, (arXiv:1510.03918)
Dan Licata, Guillaume Brunerie, A cubical approach to synthetic homotopy theory (pdf)
On 2-group-extensions of torus groups:
On the stable homotopy type of the torus:
Last revised on February 5, 2023 at 18:25:09. See the history of this page for a list of all contributions to it.