Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively.
1) how can I prove, that $nT$ and $nP$ are homeomorphic to their fundamental polygons, i.e. the quotients of the regular $4n$-gon / $2n$-gon, depicted below? I'm guessing induction on $n$. For $n=1$, the thing already holds, but how does one go through the inductive step? No need for actual formulas, just intuitively to the point when one can easily figure out the formulas himself.
2) We can see that all points become the same under the quotient projection. This gives $nT$ and $nP$ the CW-structure, depicted below,
$~~~~~~$ where we attach a disc $B^2$ via the attaching map $a_1b_1a_1^{-1}b_1^{-1}\ldots a_nb_na_n^{-1}b_n^{-1}=[a_1,b_1]\ldots[a_n,b_n]$ for $nT$, and $a_1^2\ldots a_n^2$ for $nP$.
By the theorem regarding the fundamental group of a CW-complex, it follows that $\pi_1(nT)=\langle a_1,b_1,\ldots,a_n,b_n|[a_1,b_1]\ldots[a_n,b_n]\rangle~~~~\text{and}$ $\pi_1(nP)=\langle a_1,\ldots,a_n|a_1^2\ldots a_n^2\rangle.$ Is this correct?
3) How can $nP$ and $nT$ be homeomorphic to their CW-decompositions, when at the point , the locallly $2$-euclidean property does not hold (the way I see it)?