Polynomial mod-two cohomology of Eilenberg–Mac Lane spaces

First establish the excess bookkeeping. If $I=(i_1,\dots ,i_k)$ is admissible, then $i_1=e(I)+i_2+\cdots +i_k$. Thus when $e(I)>q$, instability gives $S{q}^I{\iota }_q=0$. When $e(I)=q$, the outermost square is the top square of its input, so $S{q}^I{\iota }_q=(S{q}^{(i_2,\dots ,i_k)}{\iota }_q{)}^2\mathrm{.}$ The tail's excess is at most $e(I)$, because subtracting the tail excess gives $i_1-2i_2\ge 0$. Continue deleting heads whenever the excess remains $q$; length strictly decreases, and the empty tail has excess zero. This expresses the class uniquely as a $2^t$th power of an admissible class with excess less than $q$. Conversely, if a tail $J$ is admissible with $e(J)\le q$, prepend $q+\mathrm{\mid }J\mathrm{\mid }$. This new head is at least twice the first tail entry, since that inequality is equivalent to $e(J)\le q$. The new sequence is admissible, has excess exactly $q$, and represents the square of $S{q}^J{\iota }_q$. Iterating gives precisely all these powers. The deletion/prepending constructions are inverse on sequences; they do not rely on an unproved independence of their values.

Induct on $q$. The marked projective-space model proves the base $q=1$, where the only admissible excess-zero sequence is the empty sequence. Suppose the polynomial statement holds for $K_q$. A monomial in its polynomial generators has a unique expression as a product of distinct elements $(S{q}^J{\iota }_q{)}^{2^t},e(J)<q,t\ge 0,$ by writing each exponent in binary. Hence these classes form a simple system of generators: their distinct finite products are a vector-space basis. The preceding inverse constructions index this simple system bijectively by all admissible $I$ with $e(I)\le q$. It is degreewise finite because there are finitely many finite positive integer sequences of each fixed sum and each power has positive degree.

Apply the actual path fibration supplied by the path-loop input lemma with strict fiber $F=\mathrm{\Omega }{K}_{q+1}$. Its marked weak equivalence $h:K_q\to F$ identifies their mod-two cohomology rings. Transfer the polynomial generators and simple system from $K_q$ to $F$ by the inverse of $h^{\ast }$; this inverse commutes with squares because $h^{\ast }$ is a natural cohomology isomorphism. Write ${\iota }_q$ for the resulting marked fundamental fiber class, as in the normalized-relative-lift lemma. The normalized-relative-lift lemma gives $\delta {\iota }_q=p^{\ast }{\iota }_{q+1}$. Repeated use of the connector-compatibility lemma and square naturality gives, for every admissible $I$, $\delta (S{q}^I{\iota }_q)=p^{\ast }(S{q}^I{\iota }_{q+1})\mathrm{.}$ Therefore each simple-system element, of degree $m=q+\mathrm{\mid }I\mathrm{\mid }$, survives every earlier page and has its cohomological transgression on the exact page $m+1=q+\mathrm{\mid }I\mathrm{\mid }+1$, with representative $S{q}^I{\iota }_{q+1}$, by the relative-lifts lemma. This includes all top-square powers; no unsupported assertion about a square's “expected” differential is needed.

All hypotheses of the Borel polynomial-base theorem now hold: the total path space is contractible, the base is simply connected, the actual strict fiber cohomology ring and marking are computed by the local weak-equivalence argument of the path-loop input lemma, the simple system is degreewise finite, and all its elements have the exhibited relative lifts. That theorem identifies base cohomology as the polynomial algebra on $S{q}^I{\iota }_{q+1}$ indexed by $e(I)\le q$, which is exactly $e(I)<q+1$. This proves the induction step, simultaneously proving generation and algebraic independence. All infinite index sets are used degree by degree with finite support, and strong Serre convergence is the actual published convergence statement.

Explicit fiber boundary check. In the step with base $K_q$, the fiber is $K_{q-1}$. Its degree $2q-2$ is not dropped. By the proved full polynomial statement it consists of the degree-$2q-2$ generators (admissible sequences of operation degree $q-1$ and excess less than $q-1$) together with the single product ${\iota }_{q-1}^2$. The latter is the simple-system element $S{q}^{q-1}{\iota }_{q-1}$, of excess $q-1$. Its relative lift is $S{q}^{q-1}{\iota }_q$, so the relative-lifts lemma places its differential on page $2q-1$, from $(0,2q-2)$ to $(2q-1,0)$. Thus the previously missing fiber boundary class supplies the necessary base-degree-$2q-1$ generator. For $q=2$ this says that the square of the degree-one fiber class transgresses on page three. The polynomial induction handles this class and all other powers before any strict-range truncation is made.