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)

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)