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Jun 19 at 17:48 comment added Jose Calcines Thanks, Tyron. It seems that your argument could work well. However, wouldn't there be any problems with the part of the sequence that's not involving groups and homomorphisms?
Jun 19 at 17:46 comment added Jose Calcines Thanks, Mark. Unfortunately, the statement does not appear in these books. However, the discussion is very interesting.
Jun 19 at 10:29 comment added Tyrone To a homotopy pullback is associated the fibration sequence $\Omega C\rightarrow A\times_C^hB\rightarrow A\times B$ with the obvious maps. Look at the long exact sequence in homotopy and use the homomorphisms induced by the maps $\alpha,\beta,\gamma$ to compare the sequences coming from the different homotopy pullbacks.
Jun 19 at 10:26 comment added Mark Grant Hi Jose! Have a look at the discussion at mathoverflow.net/q/132339/8103. It's possible that the books of May-Ponto or Strom mentioned there have this statement.
S Jun 19 at 9:07 review First questions
Jun 19 at 9:15
S Jun 19 at 9:07 history asked Jose Calcines CC BY-SA 4.0