Mod-two cohomology ring of infinite real projective space

Mod-two singular homology is one-dimensional in every nonnegative degree, and $(i_n{)}_{\ast }$ is an isomorphism through degree $n$. [given, F1, F2, F3] Realize ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ as the union of the projective spaces of lines in ${\mathbb{R}}^{m+1}$ under the coordinate inclusions. For each $j$, the lines whose last nonzero coordinate is the $j$th form an open $j$-cell: scale that coordinate to $1$ to identify it with ${\mathbb{R}}^j$. Its characteristic map is the quotient of the closed upper hemisphere in $S^j$, whose equator maps into ${\mathbb{R}\mathbb{P}}^{j-1}$. Hence its closure is ${\mathbb{R}\mathbb{P}}^j$, and these characteristic maps give the standard union its CW topology, one cell in every nonnegative degree. Restriction to the first $n+1$ coordinates is therefore the subcomplex consisting of the cells through dimension $n$.

The cohomology groups and restriction maps have the corresponding description. [F4, A1, step 1.1] By [F4], $H^k({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)$ is the dual of the one-dimensional group in step 1.1, hence is ${\mathbb{F}}_2$ for every $k\ge 0$. Naturality identifies $i_n^{\ast }$ with precomposition by $(i_n{)}_{\ast }$; since the latter is an isomorphism for $k\le n$, so is the former.

Set up complementary coordinate projective subspaces after fixing the additive generators. [given, F6, step 2.1] Fix $n\ge 2$ and positive $r,s$ with $r+s=n$. Use homogeneous coordinates $x_0,\dots ,x_n$. Let $E\cong {\mathbb{R}\mathbb{P}}^r$ use $x_0,\dots ,x_r$, let $F\cong {\mathbb{R}\mathbb{P}}^s$ use $x_r,\dots ,x_n$, and put $p=E\cap F=[e_r]$, $V={\mathbb{R}\mathbb{P}}^n\setminus F$, and $W={\mathbb{R}\mathbb{P}}^n\setminus E$. Scaling $x_r,\dots ,x_n$ to zero retracts $V$ and $E\setminus \{p\}$ onto the same coordinate ${\mathbb{R}\mathbb{P}}^{r-1}$; symmetrically $W$ and $F\setminus \{p\}$ retract onto ${\mathbb{R}\mathbb{P}}^{s-1}$. Scaling only $x_r$ to zero retracts ${\mathbb{R}\mathbb{P}}^n\setminus \{p\}$ onto a coordinate ${\mathbb{R}\mathbb{P}}^{n-1}$. Each formula is well defined on projective classes, never sends a representative to zero on the stated domain, fixes its target, and depends continuously on the scaling parameter.

The following three relative-to-absolute maps are isomorphisms. [F4, F5, F6, step 1.1, step 2.1, step 3.1] $H^r({\mathbb{R}\mathbb{P}}^n,V)⟶H^r({\mathbb{R}\mathbb{P}}^n),H^s({\mathbb{R}\mathbb{P}}^n,W)⟶H^s({\mathbb{R}\mathbb{P}}^n)$ and $H^n({\mathbb{R}\mathbb{P}}^n,{\mathbb{R}\mathbb{P}}^n\setminus \{p\})⟶H^n({\mathbb{R}\mathbb{P}}^n)$ For the first map, step 3.1 and [F6] identify the relevant groups of $V$ with those of ${\mathbb{R}\mathbb{P}}^{r-1}$; step 1.1 and [F4] say that $H^{r-1}({\mathbb{R}\mathbb{P}}^n)\to H^{r-1}(V)$ is onto and $H^r(V)=0$. Exactness in [F5] gives the isomorphism. The second map is symmetric, and the last uses the punctured-space retraction in exactly the same two adjacent degrees. Naturality in [F5] and the restriction isomorphisms of step 2.1 further identify the first two relative groups with $H^r(E,E\setminus \{p\})$ and $H^s(F,F\setminus \{p\})$.

The two complementary-degree generators have nonzero top product. [F6, F7, F8, step 4.1] In the affine chart $x_r\mathrm{\ne }0$, ratios identify a neighborhood of $p$ with ${\mathbb{R}}^r\times {\mathbb{R}}^s$ and identify $E,F$ with its coordinate planes. Excision in [F6], together with contraction of the unused coordinate factor, takes the two relative generators from step 4.1 to the two coordinate local generators. Their product is nonzero by [F7]. The complements $V,W$ are open and $V\cup W={\mathbb{R}\mathbb{P}}^n\setminus \{p\}$, so [F8] transports this relative product to the top relative group and then, through the last isomorphism of step 4.1, to a nonzero product in $H^n({\mathbb{R}\mathbb{P}}^n;{\mathbb{F}}_2)$. Thus the product of the unique nonzero classes in degrees $r$ and $s$ is the unique nonzero top class.

Every finite skeleton has the truncated polynomial ring on its degree-one class. [F8, step 1.1, step 2.1, step 5.1] For $n\ge 1$, let $a_n$ be the unique nonzero class in $H^1({\mathbb{R}\mathbb{P}}^n;{\mathbb{F}}_2)$. The skeleton restriction carries $a_n$ to $a_{n-1}$ when $n>1$ by step 2.1. For $n=1$, $a_1$ is nonzero and $a_1^2=0$ for dimensional reasons. Inductively assume $1,a_{n-1},\dots ,a_{n-1}^{n-1}$ are the unique nonzero classes of the preceding skeleton. Naturality in [F8] makes $a_n^k$ restrict to $a_{n-1}^k$, hence makes it nonzero for $k<n$. Step 5.1 with $r=n-1,s=1$ then makes $a_n^n=a_n^{n-1}{a}_n$ nonzero. All higher powers vanish above dimension $n$. Hence $H^{\ast }({\mathbb{R}\mathbb{P}}^n;{\mathbb{F}}_2)={\mathbb{F}}_2[a_n]\mathrm{/}(a_n^{n+1})$, obtained here without citing a B-page example.

Finite-skeleton detection gives the infinite polynomial ring. [F8, step 2.1, step 6.1] Let $a$ be the unique nonzero element of $H^1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)$. For $n\ge 1$, step 2.1 makes $i_n^{\ast }$ an isomorphism in degree one, so it sends $a$ to $a_n$; for $n=0$ the target degree-one group is zero. For any $k\ge 1$, choose $n\ge k$. Then [F8] and step 6.1 give $i_n^{\ast }(a^k)=a_n^k\mathrm{\ne }0$. Thus $a^k$ is the unique nonzero class in degree $k$ from step 2.1. The degree-zero power is the unit. Polynomial evaluation is onto degreewise and injective because a polynomial has finitely many homogeneous terms in distinct degrees.

All boundary and choice cases are accounted for. [F1, F2, F4, F5, F6, F7, F8, A1, step 1.1, step 2.1, step 3.1, step 4.1, step 5.1, step 6.1, step 7.1] The skeleton $n=0$ is a point and restriction sends $a$ to zero, while $a^0=1$ restricts to its unit. The case $k=0$ is included, there is no largest skeleton, and every fixed power is detected on any finite skeleton of dimension at least its degree. The spaces are nonempty; zero classes remain zero under restriction. Cellular chains use all characteristic cells, and the comparison in [F2] retains arbitrary singular simplices, including degenerate ones. The product calculation uses positive $r,s$ only, and the base $n=0,1$ cases were separate. AC is used only in field duality [F4] and the local relative-product supplier [F7]; the coordinate and finite-induction arguments make no new choices. No biconditional or converse is asserted. ∎