I have several related questions, i do not know which one is more important to me, i think it would depend on their answers.
Is it true that the Euler characteristic of a finite connected aspherical simplicial 2-complex cannot be greater than 1?
If $A$ is a finite simplicial 2-complexe that retracts by deformation onto a graph (1-complex), is it true that every subcomplex of $A$ is aspherical?
(This is the question that i am most likely interested in.) If $A$ is as above, is it true that every connected subcomplex $B$ of $A$ is of Euler characteristic at most 1, and if the Euler characteristic of $B$ is 1, then $B$ is contractible?
