Summary

This is a famous example in topology called "Bing's house," named for R. H. Bing who created it. It is a house with two rooms. You enter the top room from the bottom face and the bottom room from the top face. If we view it as a 2-dimensional object (with 2-dimensional walls), Bing's house is a contractible (here's a nice sequence of illustrations showing that it is contractible), 2-dimensional simplicial complex that is not collapsible. If we view the walls having thickness, then this is topologically (homeomorphic to) a 3-ball.

To make this object, print two copies of the rooms (one for upstairs and one for downstairs) and the floor between the rooms.