homotopy

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homotopy

Etymology

From Ancient Greek ὁμός (homós, “same, similar”) + τόπος (tópos, “place”); earliest known use in print in 1922, Oswald Veblen, Analysis Situs.^[1]

Noun

homotopy (countable and uncountable, plural homotopies)

(topology) A continuous deformation of one continuous function or map to another.
The concept of homotopy represents a formalisation of the intuitive idea of a smooth deformation of one curve into another.
- 1998, Paul Sellick, “Space Exponents for Loop Spaces of Spheres”, in William G. Dwyer, editor, Stable and Unstable Homotopy, American Mathematical Society, page 279:
  An integer M is called an exponent for the torsion of an abelian group G if M * (torsion of G) = 0. We say that M is a homotopy exponent for a space X if M is an exponent for π_k (X) for all k.
- 2001, F. R. Cohen, S. Gitler, “Loop-spaces of configuration spaces, braid-like groups, and knots”, in Jaume Aguadé, Carles Broto, Carles Casacuberta, editors, Cohomological Methods in Homotopy Theory, Springer (Birkhäuser), page 63:
  A graded Lie algebra arises from these maps via the Samelson product in homotopy, the so-called homotopy Lie algebra which is discussed below.
- 2010, Vladimir G. Turaev, Homotopy Quantum Field Theory, European Mathematical Society, page xi,
  In this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d + 1)-dimensional cobordisms endowed with maps to a given space X.
(uncountable) The relationship between two continuous functions where homotopy from one to the other is evident.
(informal) Ellipsis of homotopy theory. (the systematic study of homotopies and their equivalence classes).
(topology) A theory associating a system of groups with each topological space.
(topology) A system of groups associated with a topological space.

Usage notes

Formally, there are two alternative formulations:^[2]
- Given topological spaces $X,Y$ $X,Y$ and continuous maps $f,g:X\rightarrow Y$ $f,g:X\rightarrow Y$
  1. A continuous map $H:[0,1]\times X\rightarrow Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ $\forall x\in X$ .
  2. A family of continuous maps $h_{t}:X\rightarrow Y,t\in [0,1]$ such that $h_{0}=f,$ $h_{1}=g$ and the map $(x,t)\mapsto h_{t}$ is continuous from $X$ to $Y$ . (Note that it is not sufficient to require that each map $h_{t}(x)$ be continuous.)
- Replacing the unit interval $[0,1]$ with the affine line A¹ leads to A¹ homotopy theory.
The adjective homotopic is used specifically in the sense, with respect to two functions, of "having the relationship of being in homotopy".
- Being homotopic is an equivalence relation on the class of all continuous functions between given topological spaces. An equivalence class of such a relation is called a homotopy class.