$\DeclareMathOperator{\span}{span}$ $\DeclareMathOperator{\co}{H}$ $\newcommand{\kk}{\mathbb{F}}$ $\newcommand{\qq}{\mathbb{Q}}$ $\newcommand{\zz}{\mathbb{Z}}$ $\newcommand{\rr}{\mathbb{R}}$ $\newcommand{\semi}{\hat{\chi}_2}$ $\newcommand{\ori}[1]{\textbf{(O$_{\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.

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.

| cite | improve this answer | |
  • $\begingroup$ 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. $\endgroup$ – Cihan Jun 28 at 18:03

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.