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Jul 9 at 10:20 vote accept Oleksandr Kharchenko
Jul 8 at 3:13 answer added skupers timeline score: 14
Jul 8 at 1:36 comment added Danny Ruberman You might like to look at Randall (On 4-dimensional bundle theories. Differential topology, foliations, and group actions (Rio de Janeiro, 1992), 217–233. Contemp. Math., 161, 1994) says that $\pi_3(STOP(4))$ is $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}_2$. In other words, it seems that your exact sequence at the bottom splits. There's some more information in his related publication with Schweitzer (On Foliations, Concordance Spaces, and the Smale Conjectures) in the same volume.
Jul 7 at 18:02 history edited YCor CC BY-SA 4.0
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Jul 7 at 17:55 history edited Oleksandr Kharchenko
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S Jul 7 at 17:51 review First questions
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S Jul 7 at 17:51 history asked Oleksandr Kharchenko CC BY-SA 4.0