I'm reading a proof that contains the following sentence:
Gluing a d-ball into the boundary of a pure d-dimensional complex preserves the homeomorphism type, if the intersection is a (d-1)-ball.
Here are some definitions for reference:
Also the (d-1)-cells that are contained in only one facet form the boundary of the complex.
I'd like to ask if the following picture invalidates the statement:
If the small triangle-shaped 2-ball is glued into the bigger complex and the intersection is the edge a, then the statement holds. But if the intersection is b and c, then to me it doesn't look like the homeomorphism type is preserved. The original bigger complex has a cutpoint, whereas the newly formed complex would have none.
Regardless of the answer, how do I prove that statement?