No - Ann Arbor Michigan does not impose a city income tax. You do not need to select Detroit, if you did not live or work there.
Only the following cities in the State of Michigan impose a local income tax, as of 2016:
Albion, Battle Creek, Big Rapids, Detroit, Flint, Grand Rapids, Grayling, Hamtramck, Highland Park, Hudson, Ionia, Jackson, Lansing, Lapeer, Muskegon, Muskegon Heights, Pontiac, Port Huron, Portland, Saginaw, Springfield and Walker.
Source - Michigan Department of Treasury: