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


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*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?
 A: 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. 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. 
A: About Question 3, thanks to Andreas' answer it is enough to prove that there is no acyclic 2-complex $X$ with $\pi_1(X)=D\pi_1(X)\neq\lbrace0\rbrace$.
Indeed it would mean that $\chi(B)=1 \implies$ $B$ is weakly contractible $\overset{Whitehead}{\implies}$ $B$ is contractible.
I have no examples in mind, all I can say is that making a loop into a commutator amounts to make it bound the complementary of a disk in a torus, this is how I would go for a counterexample.
