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At title goes, I read the following lemma in Bredon's book (GTM139 Topology and Geometry Page 189). But I find that I have never seen it in any other texts on algebraic topology or homological algebra. So I would like to ask the references for it.

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    $\begingroup$ To add to Paul Frost's references, this is §26, Exer. 1 in Elements of algebraic topology by Munkres, although it is left as an exercise. You can also try to prove this lemma using spectral sequences like I did, but it doesn't seem very straightforward. $\endgroup$ Aug 23, 2018 at 0:29

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See

Wall, C. T. C. "On the exactness of interlocking sequences." Enseignement math 12 (1966): 95-100

https://www.e-periodica.ch/cntmng?pid=ens-001:1966:12::36

Hardie, K. A., and K. H. Kamps. "Exact sequence interlocking and free homotopy theory." Cahiers de topologie et géométrie différentielle catégoriques 26.1 (1985): 3-31

http://www.numdam.org/article/CTGDC_1985__26_1_3_0.pdf

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