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 $2^r - 2$.



Immersing spaces in $\mathbb{R}^{n+1}$, Stiefel-Whitney classes
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


