The present article gives another proof of the following theorem original due to M. Gromov and H. B. Lawson jun. [Ann. Math. (2) 111, 423–434 (1980; Zbl 0463.53025)]:
Theorem. Let \(M^n\) be a simply connected, closed non-spin manifold of dimension \(n \geq 5\). Then \(M\) carries a positive scalar curvature metric.
The original proof of this theorem is based on the surgery theorem, also proved in [loc. cit.], and the classification of the oriented bordism ring modulo torsion \(\Omega^{\mathrm{SO}}_\ast/\mathrm{Torsion}\). For this ring, explicit generators are known and one can show that all of these carry a positive scalar curvature metric. Using surgery theory, one can show that every simply connected, closed, non-spin manifold of dimension \(n \geq 5\) can be obtained from these generators using surgeries of codimension \(\geq 3\). The surgery theorem now implies that this simply connected manifold carries a positive scalar curvature metric as well.
Führing reproves Theorem 1 in the spirit of S. Stolz [Ann. Math. (2) 136, No. 3, 511–540 (1992; Zbl 0784.53029)] by providing a different class of generators. If \(G=SU(3) \rtimes \mathbb{Z}_2\) and \(H=S(U(2)U(1)) \rtimes \mathbb{Z}_2\) then there is a universal \(\mathbb{C}P^2\)-fibre bundle with structure group \(G\): \[ \mathbb{C}P^2 = G/H \hookrightarrow BH \twoheadrightarrow BG. \] The main result of the article is the following one:
Theorem. For \(n \geq 1\), the homomorphism \[ \psi \colon \Omega^{\mathrm{SO}}_{n-4}(BG) \rightarrow \Omega^{\mathrm{SO}}_n(pt) \] that assigns to an oriented singular bordism class \([f\colon N \rightarrow BG]\) the bordism class of the total space of its pull back \([f^\ast BH]\) is surjective.
These total spaces carry a positive scalar curvature metric so one can proceed as before.
The idea is to show that \(\psi\) is induced by a map of spectra \(T \colon MSO \wedge BG_+ \rightarrow MSO\) and to study this map with tools from stable homotopy theory. More precisely, if \(U\colon MSO \rightarrow H\mathbb{Z}\) denotes the map of spectra that classifies the Thom class of the universal oriented vector bundle and if \(\widehat{MSO}\) denotes the homotopy fibre of \(U\), then \(T\) factors through \(\widehat{MSO}\). Führing proves that the induced map \(\hat{T}\colon MSO \wedge BG_+ \rightarrow \widehat{MSO}\) with the help of the Adams spectral sequence. The first step is to analyse this map on the level of homology and to prove that \(H_\ast(\hat{T})\) induces a split surjection of \(A_\ast\)-comodules, where \(A_\ast = {H\mathbb{Z}_2}_\ast H\mathbb{Z}_2\) is the dual Steenrod algebra. Next, he shows that the Adams spectral sequence degenerates on the \(E_2\)-page for algebraic reasons. Finally, he solves the extension problem by constructing enough \(\mathbb{C}P^2\)-fibre bundles that lie in \(\mathrm{im}\hat{T}_\ast \subseteq \Omega_\ast^{\mathrm{SO}}\) and generate \(\Omega_\ast^{\mathrm{SO}}/\mathrm{Torsion}\).
The paper is written with extreme care and provides details for many folklore results. After the general introduction Section 1 and an outline of the proof in Section 2, Führing presents a detailed construction of the transfer map and describes several identifications. In Section 4, the cohomological properties of the fibre bundle \(\mathbb{C}P^2 = G/H \hookrightarrow BH \twoheadrightarrow BG\) are studied, including the total Stiefel-Whitney class of the vertical tangent bundle, the action of the first Steenrod square and the description of the fibre integration map. Section 5 provides the required foundation of \(MSO\)-module spectral and the \(A_\ast\)-comodule structure of \(MSO\). Section 6 is the technical heart of the article, in which split surjectivity of \(H_\ast(\hat{T})\) is proven using methods from Hopf-algebra and their (co)module theory. Section 7 derives the Adams spectral sequence for \(MSO \wedge BG_+\) using the \(A_\ast\)-module structure from the previous section. Using a change of ring theorem, Führing gives a fairly simple resolution that allows to calculate the \(E_2\)-page. The spectral sequence collapses because \(E_2^{s,t}\) is concentrated in \((t-s)\in 4\mathbb{N}\). The extension problem is solved using the \(\mathbb{C}P^2\)-fibre bundle construction described in Section \(8\), which might be of independent interest. The final Section 9 shortly describes the proof for the analogous statement of the unoriented bordism ring that every unoriented manifold of dimension \(n\geq 2\) is (unoriented) bordant to an \(\mathbb{R}P^2\)-fibre bundle.
Due to its detailed presentation and its precise references the article is well suited for everyone who just started to study and use the Adams spectral sequence and those who wish to see how such calculations are performed.