Can anyone show me a simple example of a manifold such that $H_{1}(\mathcal{X})=0$ but $\mathcal{X}$ is not orientable? We know the contention hold for $\pi_{1}$, I am not sure if it holds for $H_{1}$.
Simple example that $H_{1}(\mathcal{X})=0$ but $\mathcal{X}$ is still not orientable.
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algebraic-topology
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0This is enough; I shall try to find one myself. Thanks. – 2012-11-04
1 Answers
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There are no such manifolds.
Obstruction to orientability is given by first Stiefel-Whitney class $w_1\in H^1(X;\mathbb Z/2\mathbb Z)$. So if $H^1(X;\mathbb Z/2\mathbb Z)=0$ manifold $X$ is always orientable.
In other words. If loop $\gamma$ is trivial as an element of $H_1$ it is always orientation-preserving.