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Where can I find references to proofs/can anyone supply me a quick proof of the following facts?

  • If the $n$-dimensional manifold $M$ can be immersed in $\mathbb{R}^{n+1}$, then each $w_i(M)$ is equal to the $i$-fold cup product $w_1(M)^i$.
  • If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, then $n$ must be of the form $2^r - 1$ or $2^r - 2$.
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    $\begingroup$ What reference for characteristic classes are you using? $\endgroup$
    – Ryan Budney
    Commented Dec 15, 2015 at 23:45

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Read Milnor Stasheff: Characteristic classes. For the first question: The total normal Stiefel Whitney class of $M$ is $1-w_1(M)$, hence the inverse is $W(M) = 1 + w_1(M) + w_1(M)^2 +... .$

For the second Question one uses that the tandent bundle $\tau$ of $RP^n$ satisfies the equality $\tau \oplus \epsilon$ is isomorphic to $n+1$ times $\gamma$, the tautological bundle. Hence $w_i(RP^n)$ the binomial coefficient choose $i$ from $n+1$ Put the two together you get the answer to the second question

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  • $\begingroup$ Dominik, you have changed the sign of $w_1(M)$. But we are over $Z_2$. $\endgroup$
    – András Szűcs
    Commented Dec 22, 2015 at 17:18

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