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Realisability problem in arrow categories. (English) Zbl 1453.55009

Let \(G\) be a group and \({\mathcal C}\) a category. The classical group realisability problem asks whether there exists an object \(X\) in \({\mathcal C}\) such that the automorphism group \(\text{Aut}_{\mathcal C}(X)\) of \(X\) is isomorphic to \(G\). The paper under review deals with the problem in the more general setting of arrow categories. By definition, the objects of the arrow category \(\text{Arr}({\mathcal C})\) associated with \({\mathcal C}\) are morphisms in \({\mathcal C}\) and a morphism \((a, b) : f_1 \to f_2\) between objects \(f_i : A_i \to B_i\) for \(i=1\) and \(2\) are morphisms \(a\) and \(b\) in \({\mathcal C}\) with \(bf_1= f_2a\). We write \(\text{Aut}_{\mathcal C}(f)\) for \(\text{Aut}_{\text{Arr}(\mathcal C)}(f)\). Then the authors ask the following:
Question 1.1. Let \(G_1\), \(G_2\) be groups, \(H \leq G_1\times G_2\), and let \({\mathcal C}\) be a given category. Is there an object \(f : A_1 \to A_2\) in \(\text{Arr}({\mathcal C})\) such that \(\text{Aut}_{\mathcal C}(f) \cong H\) and \(\text{ Aut}_{\mathcal C}(A_i) \cong G_i\) for \(i= 1,2\)?
A binary relational system \({\mathcal S}\) over a set \(I\) consists of a set \(V({\mathcal S})\) together with a family of binary relations \(R_i({\mathcal S})\) on \(V({\mathcal S})\) for \(i\in I\). A morphism \(f : {\mathcal S_1} \to {\mathcal S_1}\) in \(I{\mathcal Rel}\) the category of binary relational systems over \(I\) is a map \(f : V({\mathcal S}_1) \to V({\mathcal S}_2)\) such that \((f(v), f(w)) \in R_i({\mathcal S}_2)\) whenever \((v, w) \in R_i({\mathcal S}_1)\) for \(i \in I\).
One of the main results, Theorem 2.2, gives an affirmative answer to Question 1.1 for arbitrary groups \(G_1\), \(G_2\), \(H \leq G_1\times G_2\) and the category \(I{\mathcal Rel}\) for some \(I\). In order to construct such objects in \(I{\mathcal Rel}\), E. Goursat’s lemma [Ann. Sci. Éc. Norm. Supér. (3) 6, 9–102 (1889; JFM 21.0530.01)] which characterizes subgroups of a product of two groups is used. Applying Theorem 1.2 and ideas developed by R. Frucht [Compos. Math. 6, 239–250 (1938; JFM 64.0596.02)] and by J. de Groot [Math. Ann. 138, 80–102 (1959; Zbl 0087.37802)], the authors deduce a result of the generalized group realisability problem for the category \({\mathcal Graph}\) of simple, undirected and connected graphs.
Theorem 1.2. Let \(G_1\), \(G_2\) be groups and \(H \leq G_1\times G_2\). Then there exist \({\mathcal G}_1\), \({\mathcal G}_2\), objects in \({\mathcal Graph}\), and \(\varphi : {\mathcal G}_1 \to {\mathcal G}_2\), an object in \({\text{Arr}}({\mathcal Graph})\), such that \({\text{Aut}}_{{\mathcal Graph}}(\varphi)\cong H\) and \({\text{Aut}}_{{\mathcal Graph}}({\mathcal G}_i)\cong G_i\) for \(i = 1, 2\).
For each integer \(n \geq 1\), the authors introduce a functor \({\mathcal M}_n\) from \({\mathcal Graph}\) to the category CDGA of commutative differential graded (DG) algebras. It follows that the functor \({\mathcal M}_n\) is almost fully faithful and the image is in the full subcategory of \(n\)-connected (indeed \(30n+17\)-connected) DG algebras. This fact settles positively Question 1.1 for the category CDGA.
Theorem 1.3. Let \(G_1\), \(G_2\) be groups and \(H \leq G_1\times G_2\). For any \(n\geq 1\), there exist \(M_1\), \(M_2\), \(n\)-connected objects in \(\text{CDGA}\), and \(\varphi : M_1 \to M_2\), an object in \({\text{ Arr}}(\text{CDGA})\), such that \({\text{Aut}}_{\text{CDGA}}(\varphi)\cong H\) and \({\text{Aut}}_{\text{CDGA}}(M_i)\cong G_i\) for \(i = 1, 2\).
Let \({\mathcal Ho}Top_*\) be the homotopy category of simply-connected pointed topological spaces. Then the authors translate the result in Theorem 1.3 to that for \({\mathcal Ho}Top_*\) by means of the realization functor, which gives an equivalence between rational homotopy types of simply-connected spaces of finite type and isomorphism classes of minimal Sullivan algebras of finite type.
Corollary 1.4. Let \(G_1\), \(G_2\) be finite groups and \(H \leq G_1\times G_2\). For every \(n\geq 1\), there exist \(X_1\), \(X_2\), \(n\)-connected objects in \({\mathcal Ho}Top_*\), and \(f : X_1 \to X_2\), an object in \({\text{Arr}}({\mathcal Ho}Top_*)\), such that \({\mathcal E}(f)\cong H\) and \({\mathcal E}(X_i)\cong G_i\) for \(i = 1, 2\). Here \({\mathcal E}(X)\) denotes the group of homotopy classes of pointed self-homotopy equivalences of a pointed space \(X\) and \({\mathcal E}(f) = \text{Aut}_{{\mathcal Ho}Top_*}(f)\).
The key to proving the corollary is that the homotopy equivalence relation between the DG algebras in the image of the functor \({\mathcal M}_n\) is trivial. Corollary 4.10 in the paper describes the result.
The paper concludes with an explicit example which illustrates the construction applied in the proof of Theorem 2.2.

MSC:

55P10 Homotopy equivalences in algebraic topology
55P62 Rational homotopy theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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