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It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true in the case of odd primes. More precisely

Question: Given a smooth manifold $M$, is it true that the Pontrjagin classes $p_i\in H^*(M,\mathbb{Z}/q\mathbb{Z})$ are generated, over the algebra of Steenrod powers, by those of the form $p_{q^i}$?

In the case of the Siefel-Whiney classes, that fact can be obtained using the formula $$ Sq^i(w_j)=\sum_{t=0}^i {j+t-i-1 \choose t} w_{i-t}w_{j+t}. $$ As far as I understand, such a nice formula does not exists in the case of odd primes. Nevertheless, my question is much weaker, and I have the feeling it should be known, at least in the case $q=3$ (which I am mainly interested in).

EDIT: As Oscar points out, the answer to this question is NO for $p\geq 5$. As far as I see it, though, it is not trivially false for $p=3$.

It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true in the case of odd primes. More precisely

Question: Given a smooth manifold $M$, is it true that the Pontrjagin classes $p_i\in H^*(M,\mathbb{Z}/q\mathbb{Z})$ are generated, over the algebra of Steenrod powers, by those of the form $p_{q^i}$?

In the case of the Siefel-Whitney classes, that fact can be obtained using the formula $$ Sq^i(w_j)=\sum_{t=0}^i {j+t-i-1 \choose t} w_{i-t}w_{j+t}. $$ As far as I understand, such a nice formula does not exists in the case of odd primes. Nevertheless, my question is much weaker, and I have the feeling it should be known, at least in the case $q=3$ (which I am mainly interested in).

EDIT: As Oscar points out, the answer to this question is NO for $p\geq 5$. As far as I see it, though, it is not trivially false for $p=3$.

It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true in the case of odd primes. More precisely

Question: Given a smooth manifold $M$ and an odd prime $q$, is it true that the Pontrjagin classes $p_i\in H^*(M,\mathbb{Z}/q\mathbb{Z})$ are generated, over the algebra of Steenrod powers, by those of the form $p_{q^i}$?

In the case of the Siefel-Whiney classes, that fact can be obtained using the formula $$ Sq^i(w_j)=\sum_{t=0}^i {j+t-i-1 \choose t} w_{i-t}w_{j+t}. $$ As far as I understand, such a nice formula does not exists in the case of odd primes. Nevertheless, my question is much weaker, and I have the feeling it should be known, at least in the case $q=3$ (which I am mainly interested in).

It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true in the case of odd primes. More precisely

Question: Given a smooth manifold $M$, is it true that the Pontrjagin classes $p_i\in H^*(M,\mathbb{Z}/q\mathbb{Z})$ are generated, over the algebra of Steenrod powers, by those of the form $p_{q^i}$?

In the case of the Siefel-Whiney classes, that fact can be obtained using the formula $$ Sq^i(w_j)=\sum_{t=0}^i {j+t-i-1 \choose t} w_{i-t}w_{j+t}. $$ As far as I understand, such a nice formula does not exists in the case of odd primes. Nevertheless, my question is much weaker, and I have the feeling it should be known, at least in the case $q=3$ (which I am mainly interested in).

EDIT: As Oscar points out, the answer to this question is NO for $p\geq 5$. As far as I see it, though, it is not trivially false for $p=3$.