I have several related questions, i do not know which one is more important to me, i think it would depend on their answers.


1. Is it true that the Euler characteristic of a finite connected aspherical simplicial 2-complex cannot be greater than 1?
2. 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?
3. (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?



Is the Euler characteristic of aspherical connected 2-complexes at most 1? (No!)  What can be said about subcomplexes of 2-complexes deformation retractible onto graphs.
This answer concerns Question 1. Take a free group on $2$ generators and $k$ very long relations at random. Then, with high probability, the representation satisfies a small cancellation condition and the associated 2-complex $X$ is aspherical. Moreover, if this happens you also have $\chi(X) = k-1$.


A negative answer to Question 2 would also be a negative answer to [Whitehead's Asphericity Conjecture](http://en.wikipedia.org/wiki/Whitehead_conjecture). I believe that a positive answer is not easy to obtain either.


Concerning Question 3: If $X \subset Y$ are 2-complexes, then the induced map $H\_2(X) \to H\_2(Y)$ is injective. This follows from the long exact sequence in homology. If $Y$ deformation retracts onto a 1-complex, then this implies that $H\_2(X)=0$ and hence $\chi(X) \leq 1$ if $X$ is connected. I do not see how $\chi(X)=1$ could imply that $X$ is contractible. 


