# When does an open manifold admit two linearly independent vector fields?


Let $$M$$ be a smooth connected $$d$$-manifold (without boundary), and write $$\span(M)$$ for the maximum number of linearly independent vector fields on $$M$$. By Poincaré-Hopf, we know that if $$M$$ is closed, then $$\span(M) \geq 1$$ if and only if $$\chi(M) = 0$$. It is also known that every open $$M$$ satisfies $$\span(M) \geq 1$$. I am interested in the characterization of the condition $$\span(M) \geq 2$$. This has been achieved for closed manifolds with works of several people (over the years ~1965-90), as I will explain.

Notation for closed $$M$$: Given a field $$\kk$$, write $$b_j(M;\kk) := \dim_\kk{\co_j(M;\kk)}$$. Write $$\semi(M;\kk) := \left( \sum_{j \geq 0} b_{2j}(M;\kk) \!\!\!\mod{\!2}\right) \in \kk_2$$ for the Kervaire semi-characteristic over $$\kk$$ (it is always a mod-2 number). For each $$0 \leq j \leq d$$, write $$w_j(M) \in \co^j(M;\kk_2)$$ for the $$j$$-th Stiefel-Whitney class. Write $$[M] \in \co_d(M;\kk_2)$$ for the $$\kk_2$$-fundamental class and $$\langle-,[M] \rangle \colon \co^k(M;\kk_2) \rarr \co_{d-k}(M;\kk_2)$$ the associated PD isomorphism. If $$M$$ is oriented and $$d \equiv 0 \!\pmod{4}$$, let $$\sigma(M)$$ denote its signature.

For the characterization of closed $$M$$ with $$\span(M) \geq 2$$, first note that when $$d=2$$ we only have the $$2$$-torus. Assuming $$d \geq 3$$, then $$M$$ has $$\span(M) \geq 2$$ if and only if in addition to $$\chi(M) = 0$$, it satisfies one of the following conditions:

$$\ori{0}$$ : $$M$$ is orientable, $$d \equiv 0\!\pmod{4}$$, and $$\sigma(M) \equiv 0\!\pmod{4}$$.

$$\ori{1}$$ : $$M$$ is orientable, $$d \equiv 1\!\pmod{4}$$, $$w_{d-1}(M) = 0$$, and $$\semi(M;\rr) = 0$$

$$\ori{2,3}$$ : $$M$$ is orientable and $$d \equiv 2,3 \!\pmod{4}$$.

$$\nori{0,2}$$ : $$M$$ is non-orientable, $$d$$ is even, and writing $$\zz_{w_1(M)}$$ for the orientation sheaf, the twisted Bockstein $$\beta^{*}\colon \co^{d-2}(M;\kk_2) \rightarrow \co^{d-1}(M;\zz_{w_1(M)})$$ sends $$w_{d-2}(M)$$ to $$0$$.

$$\nori{1}$$ : $$M$$ is non-orientable, $$d \equiv 1\!\pmod{4}$$, $$w_1(M)^2 = 0 = w_{d-1}(M)$$, and $$\semi(M;\kk_2) = \langle w_2(M)w_{d-2}(M), [M] \rangle \in \kk_2 \, .$$

$$\nori{3}$$ : $$M$$ is non-orientable, $$d \equiv 3\!\pmod{4}$$, and $$w_1(M)^2 = 0$$.

$$\nori{1,3}$$ : $$M$$ is non-orientable, $$d$$ is odd, $$w_1(M)^2 \neq 0$$, and $$w_{d-1}(M) = 0$$.

[I can give precise references with a bit more work, but for now I will note that $$\ori{0}$$ is due to Frank and independently Atiyah for $$d > 4$$ and due to Randall when $$d=4$$, $$\ori{1}$$ is due to Atiyah, $$\ori{2,3}$$ is due to E. Thomas, $$\nori{0,2}$$ is due to Pollina, $$\nori{1}$$ and $$\nori{3}$$ are due to Randall, $$\nori{1,3}$$ is due to Mello.]

For open manifolds, I could only find the following: Non-orientable surfaces necessarily have $$\span = 1$$, and open orientable manifolds of dimension 2,3 are parallelizable.

Can we similarly characterize open $$d$$-manifolds $$M$$ with $$\span(M) \geq 2$$? For instance, can the method in this answer be adapted to show that $$\span(M) \geq 2$$ whenever $$M$$ is orientable and $$d \equiv 3\!\pmod{4}$$?

Throughout we assume $$d>4$$ and $$d$$ odd. Denote by $$V_{d,2}$$ the Stiefel-manifold of orthonormal $$2$$-frames in $$\mathbb R^d$$. Since $$V_{d,2}$$ is $$(d-3)$$-connected there is a $$2$$-field over the $$(d-2)$$-skeleton of $$M$$. The first obstruction to extend this $$2$$-field over the $$(d-1)$$-skeleton lies in $$H^{d-1}(M;\pi_{d-2}V_{d,2}) =H^{d-1}(M;\mathbb Z_2)$$ and is given by $$w_{d-1}(M)$$. Suppose this class vanishes and consider an extension of the $$2$$-field over the $$(d-1)$$-skeleton. But since $$M$$ is open, there are no $$n$$-cells for $$n>d-1$$. Hence the only obstruction to extend a $$2$$-field from the $$(d-2)$$-skeleton to the whole open manifold is the Stiefel-Whitney class $$w_{d-1}(M)$$.
All other obstructions in the theorems you mentioned, are coming from the existence of a $$d$$-cell of a $$d$$-dimensional closed manifold.
• I see. It is a theorem of Massey that $w_{d-1}(M) = 0$ whenever $d \equiv 3 \pmod{4}$ with $M$ is orientable and closed. I think we can drop the closed assumption by arguing that $w_{d-1}$ still vanishes because it vanishes on compact submanifolds. Thus there is always a $2$-field in this case. – Cihan Jun 28 at 18:03