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First things first, I am aware of the existence of this topic. It's related, but old and my question haven't been discussed there. So I hope it's not wrong to start a new topic ...

I'm currently searching for a vector bundle $E\to M$ with $M$ a manifold (no conditions on the rank of $E$) such that $$ w(E) = 1 + w_2(E)+w_3(E) $$ with $w_2(E),w_3(E)$ both non zero.

I've shown that :

• If $w_2(E)=0$ then $w_3(E)=0$ (Wu's formula), so we can't really simplify the question.
• $M$ can not have a $\mathbf Z/2$ cohomology engendered by a single element of degree $1$. In fact one can not have $w_2(E)=x^2$ with $x$ of degree 1.
• $E\to M$ can not be the tangent bundle of $M$ (where $M$ is in this case a smooth manifold of dimension $3$ or $4$). (Wu's formula for tangent bundle and results on spin structures in dimension 4)

If you have any ideas, It would be much appreciated. Thanks!

edit: I'd like something less "trivial" than just the universal oriented vector bundle on something like an approximation of the grassmannian. The goal is to get something like a geometrical interpretation of such a total class. It's well understood that $w_1$ represents orientation and $w_2$ spin structure, but it's still a deep mystery to me the meaning of $w_3$.

edit2 : Mark Grant gave a first answer, and thanks to him. But it seems to me that it's not clear if such a manifold $M$ exists if we ask $M$ to be low dimensional : dimension $3$ or $4$ at most. Of course it get's uglier, mostly because we can't consider the tangent bundle as I pointed out before.

First things first, I am aware of the existence of this topic. It's related, but old and my question hasn't been discussed there. So I hope it's not wrong to start a new topic.

I'm currently searching for a vector bundle $E\to M$ with $M$ a manifold (no conditions on the rank of $E$) such that $$ w(E) = 1 + w_2(E)+w_3(E) $$ with $w_2(E),w_3(E)$ both non zero.

I've shown that :

• If $w_2(E)=0$ then $w_3(E)=0$ (Wu's formula), so we can't really simplify the question.
• $M$ can not have a $\mathbf Z/2$ cohomology engendered by a single element of degree $1$. In fact one can not have $w_2(E)=x^2$ with $x$ of degree 1.
• $E\to M$ can not be the tangent bundle of $M$ (where $M$ is in this case a smooth manifold of dimension $3$ or $4$). (Wu's formula for tangent bundle and results on spin structures in dimension 4)

If you have any ideas, It would be much appreciated. Thanks!

edit: I'd like something less "trivial" than just the universal oriented vector bundle on something like an approximation of the grassmannian. The goal is to get something like a geometrical interpretation of such a total class. It's well understood that $w_1$ represents orientation and $w_2$ spin structure, but it's still a deep mystery to me the meaning of $w_3$.

edit2 : Mark Grant gave a first answer, and thanks to him. But it seems to me that it's not clear if such a manifold $M$ exists if we ask $M$ to be low dimensional : dimension $3$ or $4$ at most. Of course it get's uglier, mostly because we can't consider the tangent bundle as I pointed out before.

First things first, I am aware of the existence of this topic. It's related, but old and my question haven't been discussed there. So I hope it's not wrong to start a new topic ...

I'm currently searching for a vector bundle $E\to M$ with $M$ a manifold (no conditions on the rank of $E$) such that $$ w(E) = 1 + w_2(E)+w_3(E) $$ with $w_2(E),w_3(E)$ both non zero.

I've shown that :

• If $w_2(E)=0$ then $w_3(E)=0$ (Wu's formula), so we can't really simplify the question.
• $M$ can not have a $\mathbf Z/2$ cohomology engendered by a single element of degree $1$. In fact one can not have $w_2(E)=x^2$ with $x$ of degree 1.
• $E\to M$ can not be the tangent bundle of $M$ (where $M$ is in this case a smooth manifold). (Wu's formula for tangent bundle)

If you have any ideas, It would be much appreciated. Thanks!

edit: I'd like something less "trivial" than just the universal oriented vector bundle on something like an approximation of the grassmannian. The goal is to get something like a geometrical interpretation of such a total class. It's well understood that $w_1$ represents orientation and $w_2$ spin structure, but it's still a deep mystery to me the meaning of $w_3$.

First things first, I am aware of the existence of this topic. It's related, but old and my question haven't been discussed there. So I hope it's not wrong to start a new topic ...

I'm currently searching for a vector bundle $E\to M$ with $M$ a manifold (no conditions on the rank of $E$) such that $$ w(E) = 1 + w_2(E)+w_3(E) $$ with $w_2(E),w_3(E)$ both non zero.

I've shown that :

• If $w_2(E)=0$ then $w_3(E)=0$ (Wu's formula), so we can't really simplify the question.
• $M$ can not have a $\mathbf Z/2$ cohomology engendered by a single element of degree $1$. In fact one can not have $w_2(E)=x^2$ with $x$ of degree 1.
• $E\to M$ can not be the tangent bundle of $M$ (where $M$ is in this case a smooth manifold of dimension $3$ or $4$). (Wu's formula for tangent bundle and results on spin structures in dimension 4)

If you have any ideas, It would be much appreciated. Thanks!

edit: I'd like something less "trivial" than just the universal oriented vector bundle on something like an approximation of the grassmannian. The goal is to get something like a geometrical interpretation of such a total class. It's well understood that $w_1$ represents orientation and $w_2$ spin structure, but it's still a deep mystery to me the meaning of $w_3$.

edit2 : Mark Grant gave a first answer, and thanks to him. But it seems to me that it's not clear if such a manifold $M$ exists if we ask $M$ to be low dimensional : dimension $3$ or $4$ at most. Of course it get's uglier, mostly because we can't consider the tangent bundle as I pointed out before.

First things first, I am aware of the existence of this topic. It's related, but old and my question haven't been discussed there. So I hope it's not wrong to start a new topic ...

I'm currently searching for a vector bundle $E\to M$ with $M$ a manifold (no conditions on the rank of $E$) such that $$ w(E) = 1 + w_2(E)+w_3(E) $$ with $w_2(E),w_3(E)$ both non zero.

I've shown that :

• If $w_2(E)=0$ then $w_3(E)=0$ (Wu's formula), so we can't really simplify the question.
• $M$ can not have a $\mathbf Z/2$ cohomology engendered by a single element of degree $1$. In fact one can not have $w_2(E)=x^2$ with $x$ of degree 1.
• $E\to M$ can not be the tangent bundle of $M$ (where $M$ is in this case a smooth manifold). (Wu's formula for tangent bundle)

If you have any ideas, It would be much appreciated. Thanks!

First things first, I am aware of the existence of this topic. It's related, but old and my question haven't been discussed there. So I hope it's not wrong to start a new topic ...

I'm currently searching for a vector bundle $E\to M$ with $M$ a manifold (no conditions on the rank of $E$) such that $$ w(E) = 1 + w_2(E)+w_3(E) $$ with $w_2(E),w_3(E)$ both non zero.

I've shown that :

• If $w_2(E)=0$ then $w_3(E)=0$ (Wu's formula), so we can't really simplify the question.
• $M$ can not have a $\mathbf Z/2$ cohomology engendered by a single element of degree $1$. In fact one can not have $w_2(E)=x^2$ with $x$ of degree 1.
• $E\to M$ can not be the tangent bundle of $M$ (where $M$ is in this case a smooth manifold). (Wu's formula for tangent bundle)

If you have any ideas, It would be much appreciated. Thanks!

edit: I'd like something less "trivial" than just the universal oriented vector bundle on something like an approximation of the grassmannian. The goal is to get something like a geometrical interpretation of such a total class. It's well understood that $w_1$ represents orientation and $w_2$ spin structure, but it's still a deep mystery to me the meaning of $w_3$.