Difference between homotopy equivalence and based homotopy equivalence?

Question

As the question suggests, what is the difference between homotopy equivalence and based homotopy equivalence?

Answer 1

The difference between free/based homotopies, is that we substitute all definitions by based definitions for the latter. I.e. a homotopy between maps of pairs $f,g: (X,x)\to (Y,y)$ requires to be a map of pairs:

$$ H: (X\times I, x \times I) \to (Y,y). $$

Now for the definition of a homotopy equivalence the obvious follows (that we require to have based maps and based homotopies).

Note that the based definition is more restrictive as we require to fix the base point all the time $I$ during the homotopy. E.g. if we would not require the fundamental group to consist of based homotopy classes of loops, conjugation in there would be trivial.

Answer 2

A standard example is to take $H$ to be the Hawaian Earring, which has a wild point at $0$. The cone $CH$ on $H$ is contractible, as is any cone; but to contract it you have to move the base "up" the cone to the vertex; if the base point is taken as the wild point at the bottom of the cone, then you cannot contract $CH$ keeping that point fixed.

This question on base points is part of a more general theory involving maps of spaces $$\begin{matrix} A & \xrightarrow{g} & B \\ i \downarrow && \downarrow j\\ X & \xrightarrow{f} & Y \end{matrix} $$ where $i,j$ are inclusions and $g,f$ are homotopy equivalences. Is then $(f,g)$ a homotopy equivalence of pairs? This is true if $i,j$ are cofibrations: see 7.4.2 of Topology and Groupoids.