How to understand the spin structures on $T^d$ and $\mathbf{RP}^d$?
- We know that $H_1(T^d,\mathbb{Z}_2)=(\mathbb{Z}_2)^d$.
It looks that there are $2^d$ choices on 1-cycle of $T^d$.
But it seems like people say there are two kinds of spin structures: even and odd. How do we obtain them from $H_1(T^d,\mathbb{Z}_2)=(\mathbb{Z}_2)^d$?
- We know that $H_1(\mathbf{RP}^d,\mathbb{Z}_2)= \mathbb{Z}_2$.
Are there two kinds of spin structures for $\mathbf{RP}^d$? How to understand them?
If $\mathbf{RP}^d$ is oriented ($d$ odd), can we exchange the two spin structures?
If $\mathbf{RP}^d$ is unoriented ($d$ even), can we exchange the two spin structures?