Posted July 24, 2011
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Cole had fell in love with Elsa (notice all of the cutscenes where he had been at the Blue Room watching Elsa sing with lovestuck eyes). It is implied by the look on Cole's face the first time meeting Elsa that he had actually fallen in love with her at first sight.
As for why he got put into arson... he had been demoted because he was seen as getting to close to discovering the corruption of the department. Roy set Cole up... he waited outside Elsa's apartment & when he saw Cole enter her apartment he used the info against him. He sold then sold the info to the Mayor (with partial motivation being to cover his own corruption) of Cole cheating on his wife with Elsa (who was a german morphine addict *note that this game takes place shortly after WWII... Germans weren't exactly liked by US citizens*) to make him out to be a dishonorable & untrustworth cop (as seen in the aforementioned newspaper). They had to wait until his hearing to decide if he deserved to be fired over the "shame" that he had brought the department & in the mean time demoted him to arson (the desk of lowest importance) because they didn't want him working high profile cases during this time.
END SPOILERS
Not really a plot hole in my eyes.
Post edited July 24, 2011 by Roberttitus