Stably Free Modules over the Klein Bottle
Stably Free Modules over the Klein Bottle
Stably Free Modules over the Klein Bottle
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Introduction<br />
Constructing a <strong>Stably</strong> <strong>Free</strong> Module<br />
Applications to <strong>the</strong> <strong>Klein</strong> <strong>Bottle</strong><br />
Algebraic Complexes of G<br />
Next, let<br />
X1 = X ∨ S 2 .<br />
So, X1 has <strong>the</strong> cellular chain complex<br />
and<br />
C∗(X1) : Z〈A, S 2 〉 −→ Z〈x, y〉 −→ Z〈v〉 → 0<br />
C∗( X1) : ZG ⊕ ZG ∂2⊕0<br />
−−−−−→ ZG ⊕ ZG ∂1<br />
−→ ZG ε<br />
−→ Z −→ 0.<br />
Notice that<br />
H2( X1) = ker(∂2 ⊕ 0) = ZG.<br />
Andrew Misseldine <strong>Stably</strong> <strong>Free</strong> <strong>Modules</strong> <strong>over</strong> <strong>the</strong> <strong>Klein</strong> <strong>Bottle</strong>