I've started studying cobordism and I found difficult formalising the abelian group structure of the oriented bordism group $\Omega_n$.
The main problem I encountered is dealing with the orientation at the boundaries, and so I did some sketches to help me visualise the problem.
And to prove existence of the inverse element for any $[M]$ (denoted with $-[M]$) under the operation, we said " consider the cylinder over $M$ and bend one end over the other." From what I understood (not english mother tongue:( ) that is the situation:
But I'm not convinced due to the fact that I don't see what's changed from the first situation to the second situation, only a little bending cannot change the orientation behaviour of the boundary. Nevertheless I tried depicting the situation.
The convention I used on the orientation of the boundary is the one saying that an oriented basis of $T_p(\partial W)$ followed by an outward pointing vector is an oriented basis of $T_pW$ where $W$ is the manifold of the bordism.
Can someone clarify these passages in the proof of the existence of inverse element?