Throughout we assume $d>4$ and $d$ odd. Denote by $V_{d,2}$ the Stiefel-manifold of orthonormal $2$-frames in $\mathbb R^n$. 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.

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.

Throughout we assume $d>4$ and $d$ odd. Denote by $V_{d,2}$ the Stiefel-manifold of orthonormal $2$-frames in $\mathbb R^n$. Since $V_{d,2}$ is $(d-3)$-connected there is a $2$-field over the $(n-2)$-skeleton of $M$. The first obstruction to extend this $2$-field over the $(n-1)$-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 $(n-2)$-skeleton to the whole open manifold is the Stiefel-Whitney class $w_{n-1}(M)$.

All other obstructions in the theorems you mentioned, are coming from the existence of a $d$-cell of a $d$-dimensional closed manifold.

Throughout we assume $d>4$ and $d$ odd. Denote by $V_{d,2}$ the Stiefel-manifold of orthonormal $2$-frames in $\mathbb R^n$. 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.