Difference cochain properties

Question

I'm reading about obstruction theory. It's said that difference cochain $\delta (f_n,g_n)$ has properties:

$\delta(f_n,g_n)=0$ iff $f_n\simeq g_n (rel X_{n-1})$.

$\delta(f,g)-\delta(g,h)=\delta(f,h).$

$\delta(f,g)=-\delta(g,f).$

$d\delta(f,g)=c(g)-c(f)$.

I'm looking for book or lectures where these properties are proved. Also, detailed answers would be really helpful.

Thanks in advance.

Answer 1

In Whitehead's (1978) textbook "Elements of homotopy theory" you can find a chapter 5 "Obstruction Theory" (pp. 228 - 235).

For most of your properties you will find a proof in "Obstruction Theory: On Homotopy Classification of Maps" by H. J. Baues (1977), e.g. on pp. 261 (4.2.9 "Obstruction theorem") you will find a proof for property 1 and 4.

Also Spanier's (1966) "Algebraic topology" (pp. 269 - 276, 429 - 432) will give some hints.