Postnikov invariants of crossed complexes (Q1770477)

A crossed complex is a chain complex of groups with operators from a fixed groupoid satisfying the properties of the universal example \(\pi({\mathbf X}):\) Here \({\mathbf X} = \{ X_0 \subseteq X_1 \subseteq \cdots \}\) is a filtered space, \(\pi_1({\mathbf X}) = \pi_1(X_1, X_0)\) is the fundamental groupoid corresponding to the basepoints \(X_0\) and \(\pi_n({\mathbf X}) = \pi_n(X_n, X_{n-1})\) with the usual action of the fundamental groupoid on the higher homotopy groups. Introduced by J.~H.~C. Whitehead in his work on the classification of low-dimensional CW complexes, crossed complexes are fundamental objects for the algebraic classification of integral homotopy types (see [\textit{H.~J. Baues}, Combinatorial homotopy and \(4\)-dimensional complexes. De Gruyter Expositions in Mathematics, 2. (Berlin) etc.: Walter de Gruyter. (1991; Zbl 0716.55001)]), non-abelian homotopy theory (see [\textit{R. Brown} and \textit{P.~J. Higgins}, J. Pure and Appl. Algebra, 21, 233--260 (1981; Zbl 0468.55007]) with applications to the cohomology of groups (see [\textit{D.~F. Holt}, J. Algebra 60, No.~2, 307--320 (1979; Zbl 0699.20040)]). The category \textbf{Crs} of crossed complexes represents a reasonable first approximation to the category of topological spaces. In particular, a crossed complex admits a geometric realization which gives an adjoint to the functor \(\pi\) when restricted to CW complexes filtered by their skeleta [\textit{R. Brown} and \textit{P.~J.Higgins}, Math. Proc. Cambridge Philos. Soc. 110, 95--120 (1991; Zbl 0732.55007)]. It is natural to ask for descriptions in \textbf{Crs} of basic homotopy-theoretic constructions. In this paper, the authors provide such a description of the Postnikov tower and the \(k\)-invariants of a space. Recall the Postnikov tower of a space \(X\) is a tower \(\cdots \to P_{n+1}(X) \overset{p_{n+1}}{\longrightarrow} P_{n}{(X)} \cdots\) of principal fibrations such that the fibre of \(p_{n+1}\colon P_{n+1}(X) \to P_{n}(X)\) is the Eilenberg-Maclane space \(K(\pi_{n+1}(X), n+1)\). The fibrations \(p_{n+1}\) are thus induced by cohomology classes \(k_{n+1} \in H^{n+2}(P_{n}(X), \pi_{n+1}(X))\) called the \(k\)-invariants of \(X\). Given a crossed complex \(\mathbb C,\) the authors construct a tower of fibrations \( \cdots \to P_{n+1}(\mathbb C) \overset{\eta_{n+1}}{\longrightarrow} P_{n}(\mathbb {C)} \cdots\) in the category \textbf{Crs} such that the fibre of \(\eta_{n+1}\) has the homotopy type of \(K(\pi_{n+1}(\mathbb C), n+1)\) with homotopy groups suitably defined in the category \textbf{Crs}. They prove this is a Postnikov tower for \(\mathbb C\) in the sense that \(\mathbb C\) is recovered as the inverse limit of the tower. As the authors observe, the construction resembles the tower of categories in [\textit{H.~J. Baues}, Algebraic homotopy. Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, (1989; Zbl 0688.55001)] although the connection between the two is left open. The most intricate result of the paper concerns the identification of the \(k\)-invariants of the Postnikov tower of a crossed complex \(\mathbb C\). The authors observe that each fibration \(\eta_{n+1} \colon P_{n+1}(\mathbb C) \to P_{n}(\mathbb C)\) can be interpreted as a \(2\)-extension or \(2\)-torsor. Results of Duskin concerning the relationship between \(2\)-torsors and cotriple cohomology (which the authors helpfully review in an appendix) imply the \(\eta_{n+1}\) determine elements \(k_{n+1}\) of degree \(2\) in a certain cotriple cohomology group. Finally, the authors construct a natural degree \(n\) map from these cotriple cohomology groups to singular cohomology groups and conclude that the \(k\)-invariants of the geometric realization of a crossed complex \(\mathbb C\) are the images of the \(k\)-invariants of \(\mathbb C\). Thus the authors give a purely algebraic approach to determining the Postnikov invariants of spaces occuring as the geometric realization of a crossed complex.