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I have a question about obstruction theory extending homotopies. I'm reading Davis & Kirk's chapter 7 (Lecture Notes in Algebraic Topology).

They say they consider the problem of "finding a homotopy between $f_0,f_1:X\to Y$ extending a fixed homotopy on $A$". Here $(X,A)$ will be a relative CW-complex.

The problem is then the extension problem: we have to extend a map $f:X\times \partial I \cup A\times I\to Y$ to the whole $X\times I$.

I recall their "main theorem of obstruction theory", namely:

If $Y$ is a path-connected $n$-simple space, and $g:X_n\to Y$ is a map, then there is a cohomology class $[\theta(g)]∈H^{n+1}(X;\pi_n Y)$ which vanishes if and only if the restriction $g|_{X_{n−1}}\to Y$ extends to a map $X_{n+1}\to Y$.

(To recall: they call a space $n$-simple if the action of $\pi_1$ on $\pi_n$ is trivial).

In this scenario, they get:

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Questions:

Why is $Y$ no longer supposed to be path-connected?

Why suppose that $f_0,f_1$ agree on $A$ (and therefore to be consistent, take the homotopy rel A)? It doesn't seem to be coherent with the "extending a fixed homotopy on $A$" they write on their original problem.

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