$\DeclareMathOperator{\span}{span}$
$\DeclareMathOperator{\co}{H}$
$\newcommand{\kk}{\mathbb{F}}$
$\newcommand{\qq}{\mathbb{Q}}$
$\newcommand{\zz}{\mathbb{Z}}$
$\newcommand{\rr}{\mathbb{R}}$
$\newcommand{\semi}{\hat{\chi}\_2}$
$\newcommand{\ori}[1]{\textbf{(O$\_{\pmb{#1}}$)}}$
$\newcommand{\nori}[1]{\textbf{(NO$\_{\pmb{#1}}$)}}$
$\newcommand{\rarr}{\rightarrow}$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](https://en.wikipedia.org/wiki/Signature_(topology)).


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](https://math.stackexchange.com/a/3625363/26141),[3](https://mathoverflow.net/a/346983/21848) are parallelizable.


Can we similarly characterize open $d$-manifolds $M$ with $\span(M) \geq 2$? For instance, can the method in [this](https://mathoverflow.net/a/346983/21848) answer be adapted to show that $\span(M) \geq 2$ whenever $M$ is orientable and $d \equiv 3\!\pmod{4}$?



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


