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Classifying maps are surjective on second homology

Statement

Assume AC. Let $X$ be a path-connected CW complex and let $G = π_{1} (X, x_{0})$ . For any based classifying map $c : X \to B G$ inducing the identity on fundamental groups, the induced homomorphism $c_{*} : H_{2} (X; Z) ⟶ H_{2} (B G; Z) = H_{2} (G; Z)$ is surjective.

Statement references

McCleary, §8bis.2

Facts & Assumptions

Given: Full AC, a path-connected CW complex $X$ , a basepoint $x_{0}$ , and $G = π_{1} (X, x_{0})$ , and an arbitrary prescribed based classifying map $c : X \to B G$ inducing the identity on $π_{1}$ .

[A1]

Full AC, written AC, is The Axiom of Choice.

[L1]

Full AC implies DC: apply AC to the family of nonempty successor sets of a serial relation, choose an initial element, and iterate the resulting choice function. The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

[F1]

Under AC, $B G$ is a connected CW model with $π_{1} (B G) ≅ G$ and $π_{k} (B G) = 0$ for $k > 1$ . The classifying space of a discrete group is a K(G,1)

[F2]

The long exact sequence of a Serre fibration computes the homotopy groups of its fiber. Long exact sequence of homotopy groups of a fibration

[F3]

The homological Serre spectral sequence gives the exact five-term sequence for the homology of a fibration. Homological Serre spectral sequence, Five term exact sequence of a first quadrant homological spectral sequence

[F4]

A simply connected path-connected space has zero first homology by the degree-one Hurewicz theorem. The first Hurewicz map is abelianization

[F5]

Under AC there is a connected CW model $K (G, 1)$ . Under DC with supplied projective-resolution data, derived group homology is naturally identified with algebraic bar homology. Existence and homotopy uniqueness of Eilenberg--Mac Lane spaces, Diagonal bar coinvariants compute group homology, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

[F6]

External input: for a connected CW $K (G, 1)$ , its universal cover $T$ is a contractible free $G$ -CW complex. Its augmented cellular chains $C_{*}^{c e l l} (T) \to Z$ form a free left $Z G$ -resolution, and their coinvariants are the cellular chains of the base. Brown, §1, printed pp. 4–5 and 7, gives this construction and the resulting identification of derived group homology with the singular homology of the base. Only that comparison is imported here. Brown, §1

Proof

Given: The data above and [A1].

1.1A1L1F1F5givenconstruct

Fix the arbitrary map $c$ in the statement, rather than selecting a special classifying map. By [F1], its target is a connected CW $K (G, 1)$ , so $π_{2} (B G) = 0$ and $π_{1} (c)$ is an isomorphism. For the derived-homology comparison in [F5], supply resolutions explicitly: over $Z G$ , take the free module on the underlying set of a module $M$ , map its basis element $[m]$ to $m$ , and repeat the same free-module construction on each successive kernel. The augmentations are onto and each image is the preceding kernel, so this is an exact free, hence projective, resolution for every $M$ . This specified construction supplies the resolution convention; [A1] and [L1] supply DC.

2.1A1L1F1F5F6step 1.1constructalgebra

Apply the external comparison [F6] to this CW model $B G$ . With $T$ its universal cover, put $P_{*} = C_{*}^{c e l l} (T)$ . Choosing an oriented lift of each cell, permitted by AC, exhibits $P_{*}$ as free over $Z G$ . Contractibility makes its augmentation exact. Thus $P_{*}$ is a supplied resolution of the trivial left module; its comparison with the resolution in step 1.1 identifies the same derived group homology. Projection to $B G$ identifies $Z \otimes_{Z G} P_{*}$ with $C_{*}^{c e l l} (B G)$ , including their differentials. Taking homology, and using cellular-to-singular homology comparison as in [F6], gives $H_{n} (G; Z) ≅ H_{n} (B G; Z)$ .

2.2F1F2F4step 1.1algebra

Replace this prescribed $c$ by its mapping-path Serre fibration $p : E \to B G$ , with $E ≃ X$ and $p$ corresponding to $c$ . In [F2], the action on fiber components is transitive because $E$ is connected, and the component of the based fiber point has stabilizer all of $π_{1} (B G)$ because $π_{1} (c)$ is onto. Thus the fiber $F$ is path-connected. Since $π_{2} (B G) = 0$ and $π_{1} (c)$ is injective, the same exact sequence gives $π_{1} (F) = 1$ . Hence $H_{1} (F; Z) = 0$ by [F4].

3.1F3step 2.1step 2.2algebra∎

Apply [F3] to $F \to E \to B G$ . Since $F$ is path-connected, its $H_{0}$ local system is canonically the constant system $Z$ . The five-term sequence therefore has consecutive terms $H_{2} (E; Z) ⟶ H_{2} (B G; Z) ⟶ H_{0} (B G; H_{1} (F; Z)) .$ The last group is zero by step 2.2. Exactness makes the first map, the edge homomorphism $p_{*}$ , surjective. Under $E ≃ X$ this map is $c_{*}$ , and step 2.1 identifies its target with $H_{2} (G; Z)$ , proving the assertion.

Depends on

Used by

String-link end inclusions identify finite nilpotent peripheral quotients Theorem

Dependency tree · two levels

63 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John McCleary, Nontrivial fundamental groups, §8bis.2 (standard reference, not scraped)
Kenneth S. Brown, Lectures on the Cohomology of Groups (standard reference, not scraped)