Rational cohomology of K(Z,n) through weak CW fiber comparison

For n≥2 the actual loop fiber of the path fibration of K(Z,n) is path-connected and has Z as its only positive homotopy group, in degree n−1, by the fibration exact sequence. A weak CW approximation L→ΩK(Z,n), preserving a basepoint vertex, makes L a marked K(Z,n−1). Marked CW uniqueness gives a marked homotopy equivalence K(Z,n−1)→L. Its composite into the strict loop fiber is a weak equivalence. The published weak-equivalence/homology lemma and the rationalization lemma identify rational homology; natural field-dual evaluation identifies rational cohomology as well. Pullback preserves cup products by the published cup-product supplier, so this is an actual cohomology-ring isomorphism. A homotopy inverse on the strict fiber is unnecessary. This construction is independent of the cohomology calculation and of Schön's theorem.

Use the standard circle CW structure with one vertex and one edge. The real-line covering is a Hurewicz fibration by the covering-homotopy supplier; its contractible total, discrete fiber, and fibration exact sequence give no higher circle homotopy groups, while the published circle fundamental-group theorem marks its degree-one group as $\mathbb{Z}$. Thus $S^1$ is a marked CW $K(\mathbb{Z},1)$. CW uniqueness, sphere homology and field duality give H^(K(Z,1);Q)=Λ(x_1). Induct on n≥2 using the preceding loop comparison. Write A=H^(K(Z,n);Q). First integral Hurewicz, the rationalization lemma and field duality give A^0=Q, A^p=0 for 0<p<n and A^n=Q. The rational cohomological Serre sequence of the path fibration has E_2=A⊗H^*(K(Z,n−1);Q): each nonzero fiber degree group is Q, so no infinite-dimensional constant-coefficient identification is being assumed. Its abutment is Q in degree zero and zero elsewhere.

For even n the fiber ring is Λ(y), |y|=n−1. Only rows 0 and n−1 occur. The only possible differential is d_n and it is nonzero, because otherwise y would survive in the positive-degree contractible abutment. Its value is a nonzero scalar multiple of the normalized generator x∈A^n; rescale the rational fiber generator y so that d_n(y)=x. For each p≥0 the map A^p y→A^{p+n}, a y↦(−1)^p a x, is injective: its upper-row kernel has no incoming differential, no subsequent outgoing differential, and would survive in positive total degree. It is surjective because its bottom-row cokernel also has no remaining incoming or outgoing differential and would survive. Induction on degree, starting with A^0=Q and the initial vanishing, therefore gives A=Q[x].

For odd n the fiber ring is Q[y], |y|=n−1 even. Before page n no differential joins two occupied rows. The class y has only the possible differential d_n into A^n, and must die. Its value is a nonzero scalar multiple of the normalized generator x∈A^n; rescale the rational fiber generator y so that d_n(y)=x. Graded commutativity gives x^2=0, and the Leibniz formula gives d_n(y^k)=k y^{k−1}x. Suppose A has a nonzero class in some degree p>n; choose the least such p. A nonzero bottom-row class a∈A^p cannot be hit by d_n: its source has base degree p−n, which is zero by initial vanishing and minimality unless p−n=n, when its differential is a scalar multiple of x^2=0. A later d_r hitting a must have source base degree p−r<p and fiber degree r−1. Nonzero smaller base degrees can only be 0 or n. In base degree 0 every positive power y^k was killed injectively by d_n, since k≠0 in Q and x y^{k−1}≠0 on that page. In base degree n every x y^k is the d_n-boundary d_n(y^{k+1})/(k+1). Thus neither possible column can supply a later incoming differential. No differential leaves the bottom row. Hence a survives to the zero positive-degree abutment, a contradiction. It follows that A=Λ(x).

A natural field-dual evaluation identifies zero cohomology with zero homology: a nonzero vector is detected by a functional under AC. The degree-n homology is Q by integral first Hurewicz and the rationalization lemma. The stated low-degree homology follows. These arguments also show exactly why the weak-fiber comparison suffices for every use of the published calculation's strict-fiber interface.