nLab smooth homotopy

Definition

In differential topology, given a pair $X$, $Y$ of smooth manifolds, and a pair of parallel smooth functions $X \underoverset{f_1}{f_0}{\rightrightarrows} Y$ between them, a smooth homotopy between these is (typically defined to be) a left homotopy by a smooth function, hence a smooth function $\eta \;\colon\; X \times \mathbb{R} \longrightarrow Y$ on the product manifold of $X$ with the real line, which restricts to $f_i$ at $X \times \{i\}$, for $i \in \{0,1\} \subset \mathbb{R}$.

More generally, for $X$, $Y$ differentiable manifolds and $f_0$ and $f_1$ differentiable functions, one may ask for differentiable homotopies, all to any given order(s) of differentiability.

Properties

Continuous homotopy implies smooth homotopy

Proposition

At least if $X$ and $Y$ are closed manifolds, then a smooth homotopy exists between any pair of parallel smooth functions $(f_0, f_1)$ between them as soon as there exists an ordinary (i.e. continuous) left homotopy between their underlying continuous functions.

More generally, if $(f_0, f_1)$ are $n+2$-fold differentiable, then an ordinary continuous homotopy between them implies an $n$-fold differentiable homotopy.

(Pontrjagin 55, Thm. 8, p. 41))

More in detail:

Proposition

Let $X$ and $Y$ by smooth manifolds where $X$ may be a manifold with boundary. If continuous functions $f,g \;\colon\; X \xrightarrow{\;} Y$ are continuously homotopic relative to a closed subset $A \subset X$, then they are also smoothly homotopic relative to $A$

(Lee 2012, Thm. 10.22 (1st ed.) / Thm. 6.29 (2nd ed.))

Example

(homotopy of smooth paths relative to their endpoints)
In the case that $X \,=\, [0,1]$ is the closed interval and $Y$ any smooth manifold, then Prop. say that continuous homotopy classes of smooth paths in $Y$, relative to their endpoints are equivalent to smooth such homotopy classes. This is relevant for instance for the characterization of the fundamental groupoid and the universal cover of a smooth manifold.

Proposition

Let $X$ and $Y$ by smooth manifolds, both possibly with boundary. If continuous functions $f,g \;\colon\; X \xrightarrow{\;} Y$ are continuously homotopic, then they are also smoothly homotopic

(Lee 2012, Thm. 9.28 (2nd ed.))

Related concepts

• smooth isotopy

• smooth homotopy type

References

• Lev Pontrjagin, Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)

• John Lee, Introduction to Smooth Manifolds, Springer (2012) [doi:10.1007/978-1-4419-9982-5, book webpage, pdf]