# "winding number" of the map: $M \to S^3$ when $M$ is a non-orientable 3-manifold

Let $$M$$ be a non-orientable three dimensional manifold. I am interested in knowing how to characterize the topological properties of the map $$M\to S^3$$ where $$S^3$$ is parameterized by $$(\phi_1, \phi_2, \phi_3, \phi_4)$$ with the constraint $$\sum_{i=1}^{4} \phi_i^2=1$$ More concretely, I want to know how the integral $$\mathcal{I}[M]\equiv \frac{2}{\pi^2} \int_{M} (\epsilon^{abc} \phi_1 \partial_a \phi_2 \partial_b \phi_3 \partial_c \phi_4) \;dx dy dz\;$$ is quantized.

In particular, it is known that the integral $$\mathcal{I}[M]$$ is quantized to be an integer for any oriented $$M$$. See the post An integral map from 3-torus $$\mathbb{T}^3$$ to 3-sphere $$S^3$$ for an explanation when $$M=T^3$$. Here I am interested to know how $$\mathcal{I}[M]$$ is quantized when $$M$$ is non-orientable.

Presumably, I expect $$\mathcal{I}[M]$$ is quantized to be $$\mathcal{I}[M]\in \frac{1}{p_M}\mathbb{Z}$$ for some integer $$p_M$$ that depends on $$M$$, and I would like to know what $$p_M$$ is. More concretely,

1) When $$M=RP^2 \times S^1$$, what is $$p_{RP^2 \times S^1}$$?

2) When $$M=KB\times S^1$$, $$KB$$ is klein bottle, what is $$p_{KB}$$?

3) What is the largest possible value of $$p_{M}$$ for all possible non-orientable manifold $$M$$?

Thanks for your help!

• How do you integrate over a non-orientable manifold? Oct 5, 2018 at 3:11
• @LordSharktheUnknown I just evaluate the density $\frac{2}{\pi}(\epsilon^{abc} \phi_1 \partial_a \phi_2 \partial_b \phi_3 \partial_c \phi_4)$ and integrate over the whole manifold (or sum over the discretized manifold if you would like to think about a discrete summation). Oct 5, 2018 at 3:27
• I don't really understand the question, but one always has the mod-2 degree of a map between closed unoriented manifolds of the same dimension, an element of $\Bbb Z/2$. In fact, the Hopf degree theorem states that if $M$ is 3-dimensional, $[M, S^3] \cong H^3(M;\Bbb Z)$. When $M$ is oriented the latter group is $\Bbb Z$ (determined by the degree), but when $M$ is unorientable it is $\Bbb Z/2$.
– user98602
Oct 5, 2018 at 5:58
• @MikeMiller Thanks! Can I understand the Hopf degree theorem this way? Let M be non-orientable, and N be its double cover which is orientable. One can first calculate the above integral on N, and it should be an integer $\mathcal{I}[N]\in \mathbb{Z}$, which is the degree. Then we compute $\mathcal{I}[M]$. $\mathcal{I}[M]= \frac{1}{2}\mathcal{I}[N]$. And different lifts $M \to N$ and $M\to N'$ of the same $M$ may differ by $2$ in $\mathcal{I}[N]$, so only the fractional part of $\mathcal{I}[M]$ is well defined. This suggests $p_M=2$. Is the above understanding correct? Oct 5, 2018 at 14:04