Definition 1

Let X and Y be topological spaces and let $f$ and $g$ be continuous mappings of X into Y. A homotopy relating $f$ to $g$ is by definition a continuous mapping $\sigma : X\times \mathbf{I}\rightarrow Y$ such that, for every $x\in X$, one has $\sigma (x,0) =f(x)$ and $\sigma (x,1) =g(x)$. One says that $f$ is homotopic to $g$ if there exists a homotopy relating $f$ to $g$.

One says that $f$ is the origin and $g$ the term of the homotopy $\sigma ($cf. below, III, p. 257, remark 2).

Let A be a subset of X. One says that the homotopy $\sigma$ is fixed on A if, for every $a\in A$, the mapping $t\mapsto \sigma (a, t)$ of $\mathbf{I}$ into Y is constant. In this case, the origin and the term of $\sigma$ coincide at every point of A.

Let X and Y be topological spaces. One says that homotopies $\sigma : X\times \mathbf{I}\rightarrow Y$ and $\tau : X\times \mathbf{I}\rightarrow Y$ are juxtaposed if the term of $\sigma$ is

N. Bourbaki and Springer-Verlag Berlin Heidelberg 2 016

N.© Bourbaki, Algebraic Topology, DOI 10.1007/978-3-662-49361-8_3 229 the origin of $\tau$, in other words if one has $\sigma (x,1) =\tau (x,0)$ for every $x\in X$. In this case, the mapping $\sigma *\tau$ of $X\times \mathbf{I}$ into Y defined by

$\sigma (x,2t)$ for $0\leqslant t\leqslant 1/2$

$\tau (x,2t-1)$ for $1/2\leqslant t\leqslant 1$

is continuous (TG, I, p. 19, prop. 4) and is a homotopy relating the origin of $\sigma$ to the term of $\tau$. It is called the juxtaposed homotopy of the homotopies $\sigma$ and $\tau$.

If $\sigma : X\times \mathbf{I}\rightarrow Y$ is a homotopy, the mapping $\overline{\sigma}: X\times \mathbf{I}\rightarrow Y$ defined bythe origin of$(\sigma x, t$. One has$)\mapsto \frac{\sigma}{\sigma}(=x,1\sigma -$. If$t)\sigma$is a homotopy relating the term ofand $\tau$ are juxtaposed homotopies$\sigma$ to

of $X\times \mathbf{I}$ into Y, the homotopies $\overline{\tau}$ and $\overline{\sigma}$ are juxtaposed and one has $\overline{\sigma*\tau}=\overline{\tau}*\overline{\sigma}$.