Kodaira-Abe (1940): is every character of the one-dimensional Betti group of a compactum induced by a continuous map into a connected compact separable abelian group?

In a 1940 note, Kunihiko Kodaira and Makoto Abe attached to every compactum F a topologized one-dimensional Betti group $B^1_{\mathbb{R}/\mathbb{Z}}(F)$ built with coefficients in the circle group, and proved that a continuous map $f\colon F\to\tilde G$ into a connected compact separable abelian group $\tilde G$ is determined, up to their notion of weak homotopy, by the induced continuous homomorphism $B^1_{\mathbb{R}/\mathbb{Z}}(F)\to\tilde G$ (a character of the Betti group). They then asked the converse: is every such character realized by a mapping class of a continuous map $F\to\tilde G$? Existence was settled in their note only when the source F is itself a connected compact separable abelian group. In the later literature, group sources were strengthened by Scheffer (1972, ordinary homotopy and arbitrary compact connected group sources), solenoid-type sources are classified by Belk-Forrest via pro-groups, and the shape-theoretic Hopf classification program covers ANR targets, but none of the cited works states or resolves the surjectivity for general compactum sources. The general question therefore remains open in the literature.

Background

A compactum is a compact metric space. A connected compact separable abelian group $\tilde G$ is an abelian topological group that is compact, metrizable and connected; the basic examples beyond the circle group $S^1=\mathbb{R}/\mathbb{Z}$ and the finite-dimensional tori are the solenoids, obtained as inverse limits of circles with finite covering maps, and the general such group is a countable inverse limit of finite-dimensional tori with surjective bonding homomorphisms (Kodaira and Abe's structure theorem; equivalently, the discrete character group $X(\tilde G)$, i.e. the group of continuous homomorphisms $\tilde G\to\mathbb{R}/\mathbb{Z}$, is countable and torsion-free). By Pontryagin duality, a compact abelian group is completely determined by its discrete character group. Kodaira and Abe worked with a homology-type invariant of a compactum F with coefficients in the circle group: the one-dimensional Betti group $B^1_{\mathbb{R}/\mathbb{Z}}(F)$, a topologized group of 1-dimensional cycles that they showed satisfies $B^1_{\mathbb{R}/\mathbb{Z}}(G)\cong G$ topologically whenever G is a connected compact separable abelian group; in modern terms it plays the role of the Pontryagin dual of the first Čech cohomology $H^1(F;\mathbb{Z})$ (Čech cohomology being the cohomology theory for general spaces defined as an inverse limit over finite open covers, which is the appropriate one for compacta) with a suitable topology, so that continuous homomorphisms $B^1_{\mathbb{R}/\mathbb{Z}}(F)\to\tilde G$ correspond to homomorphisms of the character group $X(\tilde G)$ into $H^1(F;\mathbb{Z})$. Because a compact abelian group need not be an ANR (absolute neighborhood retract, i.e. locally like a polyhedron), ordinary homotopy is too fine for classifying maps into it, and Kodaira-Abe used a weak homotopy: $f\simeq g$ iff $\psi\circ f\simeq\psi\circ g$ in the usual sense for every continuous $\psi\colon\tilde G\to Q$ into any polyhedron Q, which is the 1940 ancestor of declaring two maps equal when they agree through the inverse system of tori approximating the target. The classical classification background is Bruschlinsky's theorem $[X,S^1]\cong H^1(X;\mathbb{Z})$ for compact Hausdorff X, where $[X,S^1]$ denotes the set of homotopy classes of continuous maps $X\to S^1$, and its higher-dimensional extension, Hopf's classification of maps into spheres by cohomology for spaces of bounded dimension; the shape-theoretic version of the Hopf classification (Borsuk-Holsztyński 1971, Lisitsa 1977), shape theory being the framework that compares compacta through inverse systems of polyhedral approximations rather than through maps themselves, covers maps of arbitrary compacta into polyhedral targets via Čech cohomology. Kodaira and Abe pushed the circle-target classification to all connected compact separable abelian targets: the mapping class of $f\colon F\to\tilde G$ is determined by its induced character, and the source note asks whether this classification is complete.

Problem Statement

Let F be a compactum, i.e. a compact metric space, and let $\tilde G$ be a connected compact separable abelian group (an abelian topological group that is compact, metrizable and connected). Let $B^1_{\mathbb{R}/\mathbb{Z}}(F)$ be Kodaira-Abe's topologized one-dimensional Betti group of F with coefficients in the circle group $\mathbb{R}/\mathbb{Z}$, and call a continuous homomorphism $B^1_{\mathbb{R}/\mathbb{Z}}(F)\to\tilde G$ a character of the Betti group. Say that two continuous maps $f,g\colon F\to\tilde G$ are weakly homotopic if $\psi\circ f\simeq\psi\circ g$ in the ordinary sense for every continuous $\psi\colon\tilde G\to Q$ into any polyhedron Q. Every continuous $f\colon F\to\tilde G$ induces a character of $B^1_{\mathbb{R}/\mathbb{Z}}(F)$, and Kodaira-Abe proved that this assignment is injective on weak homotopy classes.

Question (Kodaira-Abe 1940): is the assignment surjective, i.e. is every character of $B^1_{\mathbb{R}/\mathbb{Z}}(F)$ induced by some continuous map $F\to\tilde G$, unique up to weak homotopy?

In modern terms: identifying $B^1_{\mathbb{R}/\mathbb{Z}}(F)$ with the Pontryagin dual of the first Čech cohomology $H^1(F;\mathbb{Z})$, writing $X(\tilde G)$ for the discrete character group of $\tilde G$, and writing $\tilde G=\varprojlim_i T_i$ as a countable inverse limit of finite-dimensional tori with surjective bonding homomorphisms, the question asks whether the natural map

$$[F,\tilde G]_{\mathrm{weak}}\to\mathrm{Hom}\big(X(\tilde G),\,H^1(F;\mathbb{Z})\big)\cong\varprojlim_i[F,T_i]$$

where $[F,\tilde G]_{\mathrm{weak}}$ is the set of weak homotopy classes and $[F,T_i]$ denotes ordinary homotopy classes of maps $F\to T_i$, is surjective; equivalently, whether every family of homotopy classes $F\to T_i$ that is compatible through the tower is realized by an actual continuous map $F\to\tilde G$. The source settles only the case where F is itself a connected compact separable abelian group.

References

1. Kunihiko Kodaira and Makoto Abe, 'Über zusammenhängende kompakte abelsche Gruppen', Proceedings of the Imperial Academy of Japan 16 (1940), no. 5, 167-172. DOI: 10.3792/pia/1195579160, https://doi.org/10.3792/pia/1195579160 (reprinted as item (7), pp. 26-31, in Kunihiko Kodaira: Collected Works, Volume I, DOI: 10.1515/9781400869855-008, https://doi.org/10.1515/9781400869855-008)
2. Jahrbuch über die Fortschritte der Mathematik, review by T. Tannaka of Kodaira-Abe (1940), JFM 66.0960.01, https://zbmath.org/2508056
3. Wladimiro Scheffer, 'Maps between topological groups that are homotopic to homomorphisms', Proceedings of the American Mathematical Society 33 (1972), no. 2, 562-567. DOI: 10.1090/S0002-9939-1972-0301130-8, https://doi.org/10.1090/S0002-9939-1972-0301130-8
4. James Belk and Bradley Forrest, 'Compact aspherical solenoids', arXiv:1009.5716 (2010), https://arxiv.org/abs/1009.5716
5. Edgar E. Enochs, 'Homotopy groups of compact Abelian groups', Proceedings of the American Mathematical Society 15 (1964), no. 6, 878-881. DOI: 10.1090/S0002-9939-1964-0169240-1, https://doi.org/10.1090/S0002-9939-1964-0169240-1
6. Karol Borsuk and Włodzimierz Holsztyński, 'Hopf classification theorem in the shape theory', Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 19 (1971), 387-390. Zbl 0213.49903, https://zbmath.org/3339137
7. Yu. T. Lisitsa, 'A Hopf classification theorem in shape theory', Siberian Mathematical Journal 18 (1977), no. 1, 107-119. DOI: 10.1007/BF00966955, https://doi.org/10.1007/BF00966955
8. Martin Huber and Willi Meier, 'Linearly compact groups', Journal of Pure and Applied Algebra 16 (1980), no. 2, 167-182. DOI: 10.1016/0022-4049(80)90014-6, https://doi.org/10.1016/0022-4049(80)90014-6
9. Sergey A. Melikhov, 'Steenrod homotopy', arXiv:0812.1407 (2008), https://arxiv.org/abs/0812.1407