This is an instantiation of an image on page 52 of Allen Hatcher’s Algebraic Topology. (You can download this
book for free.) Here’s how to construct it: Start with the Cartesian product Y × I. Now rotate one end by 1/3 and glue it to the other end. The resulting object has only one boundary and is not a manifold.
There is a companion project that attempts to instantiate the example on p. 52 of Hatcher, in which a disk is glued to along the boundary of the twisted/glued Y x I. It’s pictured in the last photo.