I've recently been teaching myself algebraic topology from Croom's Basic Concepts of Algebraic Topology, and I'm really lost on a problem concerning the Euler characteristic of a pseudomanifold. The problem is as follows:
Prove that if $K$ is a 2-pseudomanifold, then $\chi(K)\leq 2$, where $\chi(K)$ is the Euler characteristic of $K$.
I can see how this intuitively makes sense, but I really am not sure of where to start or how to write a formal proof of this. Any help is appreciated!
Hint: You can compute $\chi(K)$ as the alternating sum of the ranks of the homology groups of $K$, which in this case go up to only dimension $2$. So $\chi(K)=\operatorname{rank} H_0(K)-\operatorname{rank} H_1(K)+\operatorname{rank} H_2(K)$. What possible values can each of the numbers appearing in this sum take? (Don't worry if you don't know exactly what all of them are; remember that you only need to prove that $\chi(K)\leq 2$.)