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Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id X and f ∘ g is homotopic to id Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type.
Homotopical algebraic geometry: The main idea is to extend schemes by formally replacing the rings with any kind of "homotopy-ring-like object". More precisely this object is a commutative monoid in a symmetric monoidal category endowed with a notion of equivalences which are understood as "up-to-homotopy monoid" (e.g. E ∞-rings). 2002: Peter ...
The Thom isomorphism brings cobordism of manifolds into the ambit of homotopy theory. 1952: Edwin E. Moise: Moise's theorem established that a 3-dimension compact connected topological manifold is a PL manifold (earlier terminology "combinatorial manifold"), having a unique PL structure. In particular it is triangulable. [37]
In general, every manifold has the homotopy type of a CW complex; [3] in fact, Morse theory implies that a compact manifold has the homotopy type of a finite CW complex. [citation needed] Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves (or timelike multiply connected ).
The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. This curve has total curvature 6π, and turning number 3.. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if ...
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, ...