Real projective space Stiefel-Whitney non-immersion obstruction

The two displayed computations are [F1] and [F2] read in the ring ${\mathbb{F}}_2[a]\mathrm{/}(a^{m+1})$: $w(T{\mathbb{R}\mathbb{P}}^m)=(1+a{)}^{m+1}$, and $\bar{w}({\mathbb{R}\mathbb{P}}^m)=(1+a{)}^{-(m+1)}={\sum }_{i\in S_m}{a}^i$ with $d(m)=\max S_m$ the highest index with nonzero coefficient, so the highest nonzero normal class is ${\bar{w}}_{d(m)}({\mathbb{R}\mathbb{P}}^m)=a^{d(m)}\mathrm{\ne }0$ by [F4]. By definition of $d(m)$, every ${\bar{w}}_i$ with $i>d(m)$ vanishes.

For clause (i): if $k<d(m)$ then $i:=d(m)>k$ and ${\bar{w}}_i({\mathbb{R}\mathbb{P}}^m)\mathrm{\ne }0$, so [F3] forbids an immersion of ${\mathbb{R}\mathbb{P}}^m$ into ${\mathbb{R}}^{m+k}$; equivalently every immersion has codimension at least $d(m)$. For clause (ii): if $m=2^p$ with $p\ge 1$, then the binary expansion of $m$ has the single nonzero digit $2^p$, so $i\wedge m=0$ for $0\le i\le m$ exactly when $i<2^p=m$, that is $S_m=\{0,1,\dots ,m-1\}$ and $d(m)=m-1$; hence ${\bar{w}}_{m-1}=a^{m-1}\mathrm{\ne }0$ and, applying clause (i) with $k=m-2<d(m)$, there is no immersion into ${\mathbb{R}}^{m+(m-2)}={\mathbb{R}}^{2m-2}={\mathbb{R}}^{2^{p+1}-2}$. For $m=2$ this is $k=0$: the final clause of [F3] with the nonzero degree-one normal class excludes this equal-dimensional immersion; its proof treats that instance by the local-diffeomorphism and compact-image argument.

For clause (iii): write $m+1={\sum }_{j\in B}{2}^j$ with $B$ the set of binary digit positions. In characteristic two, $(1+a{)}^{m+1}={\prod }_{j\in B}(1+a^{2^j})$: this follows by iterating $(1+a{)}^{2^{j+1}}=((1+a{)}^{2^j}{)}^2=1+a^{2^{j+1}}$, as in [F2]. If $B=\{j_0\}$ has one element then $m+1=2^{j_0}$ is a power of two and $(1+a{)}^{m+1}=1+a^{m+1}=1$ in ${\mathbb{F}}_2[a]\mathrm{/}(a^{m+1})$. If $B$ has at least two elements and $j_0=\min B$, then $2^{j_0}<m+1$ and the product contains the monomial $a^{2^{j_0}}$ with coefficient $1$ (choose the factor $j_0$ and the constant term from every other factor); no other selection of factors contributes to degree $2^{j_0}$, and $2^{j_0}\le m$ so this term is nonzero in the truncation. Hence $w(T{\mathbb{R}\mathbb{P}}^m)\mathrm{\ne }1$ in that case, establishing the equivalence. The criterion concerns the tangent class only and says nothing about parallelizability or about Massey-type improvements of the non-immersion bound for general $m$.