This question regards a particular exercise regarding Algebraic Topology, CW complexes and homotopy.
I am trying to prove that the Klein Bottle is homotopic to $S^2 \vee S^1 \vee S^1 $, where $\vee $ denotes the disjoint union attaching one single point of each space. Recall that the Klein Bottle can be obtained by identifying the sides of a square by $abab^{-1}$.
So, it seems for me that the only general homotopic invariant operations that I can do over a CW complex is to quotient the space by any closed subcomplex that is nullhomotopic, or to rearrange the attaching functions on the complex homotopically. This is described with some depth in Allen Hatcher's book "Algebraic Topology", chapter 0. This is in fact a related question from the exercises, so that material should be sufficient.
No particular homotopy arrise when trying to find it by some naive approach.
Any help.
This is false. These two spaces have different fundamental groups. By Van Kampen's theorem, $\pi_1(S^2 \vee S^1 \vee S^1)$ is the free group on two generators. On the other hand, the fundamental group of the Klein bottle is $\langle a, b \mid abab^{-1}\rangle$.