2
$\begingroup$

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

$\endgroup$
2
12
$\begingroup$

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 $H_n(G,G\setminus\{g\})$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.