Green’s Dictionary of Slang

armpit n.

[the link of the SE armpit with dirt and smell]
[1960s+]

1. (orig. US) the least appetizing, poorest, most run-down and poss. dangerous area of a city or town; often as armpit of the nation/universe; also in phr. rough as a docker’s armpit.

2. (orig. US) used similarly of a place, e.g. a nightclub or café.

3. (US campus) an unpleasant individual.

In phrases

put someone up to their armpits (v.)

[mid-19C] (UK Und.) to cheat an accomplice out of his share of the proceeds.

SE in slang uses

In phrases

up to one’s armpits (also up to one’s diddies, ...one’s nipples, ...one’s tits, over one’s tits) [var. on up to the elbows under elbow n.1 ]

[19C+] (orig. US) consumed by, overwhelmed by.

work under the armpits (v.) [SE work/work v. (4); ? above the armpits is the neck from which one can be hanged, so below this fig. line one is ‘safe’]

[early 19C] (UK Und.) to confine one’s criminality to such activities that would be classed as petty larceny, bringing a maximum sentence of seven years transportation, rather than hanging; thus work above the armpits v., to commit crimes that could lead to one’s execution.