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Let h* be a multiplicative generalized cohomology theory and $E \rightarrow X$ a real vector bundle.

1. Is it true that, if $E$ is orientable with respect to h*, then it is also orientable with respect to the singular cohomology with coefficients in $h^{0}(pt)$, for $(pt)$ a space with one point?

2. Is it true that, if $E$ is orientable with respect to the singular cohomology with coefficeints in $\mathbb{Z}_{p}$, for $p > 2$, then it is also orientable with respect to the singular cohomology with coefficients in $\mathbb{Z}$?

Added later: I thought to a possible simple answer using weak orientability, actually proving what stated in the last comment. I wrote this as a comment to answer 3.

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# Orientation and generalized cohomology theories

Let h* be a multiplicative generalized cohomology theory and $E \rightarrow X$ a real vector bundle.

1. Is it true that, if $E$ is orientable with respect to h*, then it is also orientable with respect to the singular cohomology with coefficients in $h^{0}(pt)$, for $(pt)$ a space with one point?

2. Is it true that, if $E$ is orientable with respect to the singular cohomology with coefficeints in $\mathbb{Z}_{p}$, for $p > 2$, then it is also orientable with respect to the singular cohomology with coefficients in $\mathbb{Z}$?