An alternative proof of a weak form of Serre's theorem (Q1608103)

This paper gives a proof of Serre's theorem: the finiteness of stable homotopy groups of spheres. The author works on the category \({\mathcal P}\) of polyhedra. Put \(A={\mathcal P}(X,Y)\) for \(XY\in {\mathcal P}\). He calls a mapping \(f\) from \(A\) to an abelian group \(Q\) functional, and gives an order \(\prec\) of them by: \((f':A\to Q') \prec(f:A\to Q)\) if \(\sum^n_{k=1} i_kf(a_k) =0\) implies \(\sum^n_{k=1} i_kf'(a_k) =0\) for any \(n\in \mathbb{N}\), \(i_k\in \mathbb{Z}\) and \(a_k\in A\). \(f'\) is called subordinate to \(f\). The typical functionals are \(i\): \(A\to \text{Map}(X\times Y,\mathbb{Z})\) given by \(i(a) (x,y)=1\) if \(a(x)=y\), and \(=0\) otherwise, \(h\): \(A\to\Hom (H_*(X)\), \(H_*(Y))\) by \(h(a)=a_*\), and \(q\): \(A\to\Hom (P(Y), P(X))/ \text{Tors}\) by \(q(a)=a^*\) for a suitable contravariant functor \(P\) from the category of compact polyhedral pairs to the category of abelian groups. The author shows that \(f\prec h\) for a homotopy invariant functional \(f: A\to Q\) with \(f\prec i\) and \(Q\) torsion free, and that \(q\prec i\). The main tool to show these is ordinary homology theory. It follows that if \(q\) is homotopy invariant, then \(q\prec h\). Now Serre's theorem follows from the case where \(P\) is the unreduced stable cohomotopy \(\pi^*_S\). Indeed, \(h(a)=0\) for \(a:S^k\to S^{k+n}\) and \(q\prec h\).