Cell decomposition of connected sum of Torus and Klein bottle

Question

Let $K$ be the Klein bottle and $T^2$ the Torus

What is the cell composition of $K \# T^2$?

Answer 1

There are many cell decompositions for this.

One reasonable way might be to take the standard square quotients for each. The Klein bottle is given by the word $ab^{-1}ab$, and the torus is given by $cdc^{-1}d^{-1}$. Cut along the corner of each, and glue together the words. This gives the quotient of a polygon whose relation is exactly $$ab^{-1}abcdc^{-1}d^{-1}.$$

After doing the specified gluings, we can see that this is a CW-complex with $1$ zero cell, $4$ one cells, and a single two cell.

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On the other hand, if you have seen that that the klein bottle is the connect sum of two $\mathbb R P^2$'s, then we can rewrite the question in the following way:

$$T^2 \# K=(T^2 \# \mathbb RP^2) \#\mathbb RP^2=\mathbb RP^2 \#\mathbb RP^2\#\mathbb RP^2\#\mathbb RP^2=K \# K $$

where the second equality comes from associativity and the decomposition o $T^2 \# \mathbb RP^2$ into three projective planes (Theorem 2.1 here) and the final one comes from the associativity of connect sum.