Presentations of algebras and the Whitehead \(\varGamma\)-functor for chain algebras (Q1188129)

Let \((A,d)\) be a chain algebra over a principal ideal domain \(R\). Choosing a cofibrant model \((T(V),d)\rightarrow(A,d)\) one can define the \(\Gamma\)-modules \(\Gamma_n(A) := \text{im}(H_n(X^{n-1})\rightarrow H_n(X^n))\) where \(X^n = T(V_{\leq n})\). Let \(H:=H_0(A,d)\). Then there is the analogue of J.H.C. Whitehead's certain exact sequence \[ \dots \text{Tor}^A_{n+1}(H,H)\rightarrow\Gamma_{n- 1}(A)\rightarrow H_{n-1}(A)\rightarrow\text{Tor}^A_n(H,H)\dots \] Assume that \(H_j(A)=0\) for \(1\leq j\leq n-1\). It is shown that then \(\Gamma_i(A)=0\) for \(i<2n\) and \(\Gamma_{2n}(A)\approx H_n(A)\otimes_H H_n(A)\) (as \(H\)-bimodules). Moreover, there is a Künneth type spectral sequence, which converges to \(\Gamma_i(A)\) in the metastable range \(i\leq 3n-1\). Assume now that \(A=T(V)\) is free. Then the canonial map \(A\rightarrow H\) is called an \(n\)-presentation of \(H\), if \(V\) is a graded free \(R\)-module with \(V_j=0\) for \(j>n\) and if \(H_j(A)=0\) for \(1\leq j\leq n-1\). Under these conditions it is shown that \(H_n := H_n(A)\) is a projective \(H\)-module and, moreover one has \(H_\ast(A,d)\approx T_H(H_n)\), where \(T_H(H_n)=\sum_i T_H^i(H_n)\) with \(T_H^0(H_n)=H\) and \(T_H^i(H_n) = H_n\otimes_H H_n\otimes_H\dots\otimes_H H_n \) (\(i\) times). There is an analogue of \textit{J. H. C. Whitehead}'s result on trees of homotopy types [Am. J. Math. 72, 1-57 (1950; Zbl 0040.38901)]: If \(H\) is finitely generated (as an \(R\)-module), then the homotopy types of \(n\)-presentations of \(H\) form the vertices of a tree, two vertices \(X^n\) and \(Y^n\) being connected by an edge if \(Y^n\) has the same homotopy type as \(X^n\ast T(s^n)\). Here \(\ast\) denotes the free product and \(s^n\) is a generator of degree \(n\). As a topological application, the authors study the Pontryagin algebra \(H_\ast(\Omega X;R)\) of a 1-connected 2-cone \(X\), i.e. \(X\) is the mapping cone of a map \(W_1\rightarrow W_0\), where \(W_0,W_1\) are wedges of spheres. Let \(H=\text{im}(H_\ast(\Omega W_0;R)\rightarrow H_\ast(\Omega X;R))\). If \(H\) is \(R\)-free, then there is a free \(R\)-module \(V\) such that \(T(V)\otimes H\approx H_\ast(\Omega X;R)\) (as \(R\)-modules).