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Cross sections and pseudo-homomorphisms of topological abelian groups. (English) Zbl 1372.22002
Summary: We say that a mapping \(\omega\) between two topological abelian groups \(G\) and \(H\) is a pseudo-homomorphism if the associated map \((x, y) \in G \times G \mapsto \omega(x + y) - \omega(x) - \omega(y) \in H\) is continuous. This notion appears naturally in connection with cross sections (continuous right inverses for quotient mappings): given an algebraically splitting, closed subgroup \(H\) of a topological group \(X\) such that the projection \(\pi : X \rightarrow X / H\) admits a cross section, one obtains a pseudo-homomorphism of \(X / H\) to \(H\), and conversely. We show that \(H\) splits as a topological subgroup if and only if the corresponding pseudo-homomorphism can be decomposed as a sum of a homomorphism and a continuous mapping. We also prove that pseudo-homomorphisms between Polish groups satisfy the closed graph theorem. Several examples are given.

22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)
54C65 Selections in general topology
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
Full Text: DOI
[1] Armacost, D. L., The structure of locally compact abelian groups, Pure Appl. Math., vol. 68, (1981), Marcel Dekker, Inc. New York and Basel · Zbl 0509.22003
[2] Bello, H. J.; Chasco, M. J.; Domínguez, X.; Tkachenko, M., Splittings and cross sections in topological groups, J. Math. Anal. Appl., 435, 1607-1622, (2016) · Zbl 1329.22003
[3] Bessaga, C.; Pelczynski, A., Selected topics in infinite-dimensional topology, Mathematical Monographs, vol. 58, (1975), PWN Warsaw · Zbl 0304.57001
[4] Cabello, F., Quasi-homomorphisms, Fundam. Math., 178, 3, 255-270, (2003) · Zbl 1051.39032
[5] Comfort, W. W.; Saks, V., Countably compact groups and finest totally bounded topologies, Pac. J. Math., 49, 1, 33-44, (1973) · Zbl 0271.22001
[6] Comfort, W. W.; Hernández, S.; Trigos-Arrieta, F. J., Cross sections and homeomorphism classes of abelian groups equipped with the Bohr topology, Topol. Appl., 115, 2, 215-233, (2001) · Zbl 0986.22001
[7] Dikranjan, D., A class of abelian groups defined by continuous cross sections in the Bohr topology, Rocky Mt. J. Math., 32, 4, 1331-1355, (2002) · Zbl 1036.22001
[8] Engelking, R., General topology, (1989), Heldermann Verlag Berlin · Zbl 0684.54001
[9] Griffith, P., A solution to the splitting mixed group problem of Baer, Trans. Am. Math. Soc., 139, 261-269, (1969) · Zbl 0194.05301
[10] Hewitt, E.; Ross, K. A., Abstract harmonic analysis. vol. I: structure of topological groups, integration theory, group representations, Grundlehren der Mathematischen Wissenschaften, vol. 115, (1979), Springer-Verlag Berlin-New York · Zbl 0115.10603
[11] Husain, T., Introduction to topological groups, Saunders Mathematics Books, (1966), W. B. Saunders Company Philadelphia, London · Zbl 0136.29402
[12] Kalton, N. J.; Peck, N. T.; Roberts, J. W., An F-space sampler, London Mathematical Society Lecture Note Series, vol. 89, (1984), Cambridge Univ. Press Cambridge · Zbl 0556.46002
[13] Klee, V. L., Invariant metrics in groups (solution of a problem of Banach), Proc. Am. Math. Soc., 3, 484-487, (1952) · Zbl 0047.02902
[14] Trigos-Arrieta, F. J., Continuity, boundedness, connectedness and the Lindelöf property for topological groups, J. Pure Appl. Algebra, 70, 199-210, (1991) · Zbl 0724.22003
[15] Zhong, S.; Li, R.; Won, S. Y., An improvement of a recent closed graph theorem, Topol. Appl., 155, 15, 1726-1729, (2008) · Zbl 1163.46002
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