Is there an example of a non-orientable group manifold?

Question

Basically what I'm looking for is a topological group that is also a non-orientable, n-dimensional manifold

Answer 1

You can't do it. Suppose $G$ were an $n$-dimensional manifold which is a topological group. Recall that an orientation of a topological manifold $M$ is a consistent choice of generator for $H_n(M,M\setminus\{x\})\cong \mathbb Z$ for each $x\in M$. But the left-multiplication homeomorphisms $\ell_g\colon G\to G$, $x\mapsto gx$ give canonical isomorphisms from $H_n(G,G\setminus\{e\})$ to $H_n(G,G\setminus\{g\})$.