Strong homology is not additive

Mardešić, S.; Prasolov, A. V.

doi:10.1090/S0002-9947-1988-0940224-7

Strong homology is not additive
HTML articles powered by AMS MathViewer

by S. Mardešić and A. V. Prasolov

Trans. Amer. Math. Soc. 307 (1988), 725-744

DOI: https://doi.org/10.1090/S0002-9947-1988-0940224-7

PDF | Request permission

Abstract:

Using the continuum hypothesis (CH) we show that strong homology groups $\overline {{H_p}} (X)$ do not satisfy Milnor’s additivity axiom. Moreover, CH implies that strong homology does not have compact supports and that $\overline {{H_p}} (X)$ need not vanish for $p < 0$.

References

A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
Jean-Marc Cordier, Homologie de Steenrod-Sitnikov et limite homotopique algébrique, Manuscripta Math. 59 (1987), no. 1, 35–52 (French, with English summary). MR 901248, DOI 10.1007/BF01171263
C. U. Jensen, Les foncteurs dérivés de $\underleftarrow {\mmlToken {mi}{lim}}$ et leurs applications en théorie des modules, Lecture Notes in Mathematics, Vol. 254, Springer-Verlag, Berlin-New York, 1972. MR 0407091, DOI 10.1007/BFb0058395
Yu. T. Lisitsa, Duality theorems and dual categories of shapes and coshapes, Dokl. Akad. Nauk SSSR 263 (1982), no. 3, 532–536 (Russian). MR 650357
Ju. T. Lisica and S. Mardešić, Steenrod-Sitnikov homology for arbitrary spaces, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 207–210. MR 707958, DOI 10.1090/S0273-0979-1983-15156-X
Ju. T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glas. Mat. Ser. III 19(39) (1984), no. 2, 335–399 (English, with Serbo-Croatian summary). MR 790021
Juriĭ T. Lisica and Sibe Mardešić, Strong homology of inverse systems of spaces. I, Topology Appl. 19 (1985), no. 1, 29–43. MR 786079, DOI 10.1016/0166-8641(85)90083-5
Juriĭ T. Lisica and Sibe Mardešić, Strong homology of inverse systems of spaces. I, Topology Appl. 19 (1985), no. 1, 29–43. MR 786079, DOI 10.1016/0166-8641(85)90083-5
Juriĭ T. Lisica and Sibe Mardešić, Strong homology of inverse systems of spaces. I, Topology Appl. 19 (1985), no. 1, 29–43. MR 786079, DOI 10.1016/0166-8641(85)90083-5
Yu. T. Lisitsa and S. Mardešić, Steenrod homology, Geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986, pp. 329–343. MR 925874
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
Sibe Mardešić, Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math. 114 (1981), no. 1, 53–78. MR 643305, DOI 10.4064/fm-114-1-53-78
Sibe Mardešić, A note on strong homology of inverse systems, Tsukuba J. Math. 11 (1987), no. 1, 177–197. MR 899730, DOI 10.21099/tkbjm/1496160511
Sibe Mardešić and Jack Segal, Shape theory, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982. The inverse system approach. MR 676973
S. Mardešić and T. Watanabe, Strong homology and dimension, Topology Appl. 29 (1988), no. 2, 185–205. MR 949368, DOI 10.1016/0166-8641(88)90074-0
William S. Massey, Homology and cohomology theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., New York-Basel, 1978. An approach based on Alexander-Spanier cochains. MR 0488016
L. D. Mdzinarishvili, On total homology, Geometric and algebraic topology, Banach Center Publ., vol. 18, PWN, Warsaw, 1986, pp. 345–361. MR 925875
J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341. MR 159327, DOI 10.2140/pjm.1962.12.337
Z. R. Miminoshvili, The exact and semi-exact homology sequence of arbitrary spaces, Soobshch. Akad. Nauk Gruzin. SSR 113 (1984), no. 1, 41–44 (Russian, with English and Georgian summaries). MR 780701
T. Watanabe, Čech homology, Steenrod homology and strong homology. I, Glas. Mat. Ser. III 22(42) (1987), no. 1, 187–238 (English, with Serbo-Croatian summary). MR 940106