The Thom isomorphism for nonorientable bundles

E. G. Sklyarenko

Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 4, pp. 55-103.

Résumé

The classical theory of Thom isomorphisms is extended to nonorientable vector bundles. The properties of orientation sheaves of bundles and of the Thom and Euler classes $\tau$ and $e$ with respect to projections, fiber maps, Cartesian products, and Whitney sums of bundles are studied. The validity of standard constructions used in the applications of the classes $\tau$ and $e$ is confirmed. It is shown that the Thom isomorphisms, together with their form, are consequences of the Poincaré duality.

Bibliographie

[1] Bott R., Tu L. V., Differentsialnye formy v algebraicheskoi topologii, Nauka, M., 1988 | MR

[2] Burbaki N., Obschaya topologiya. Osnovnye struktury, Nauka, M., 1968 | MR

[3] Godeman R., Algebraicheskaya topologiya i teoriya puchkov, IL, M., 1961

[4] Grotendik A., O nekotorykh voprosakh gomologicheskoi algebry, IL, M., 1961

[5] Dold A., Lektsii po algebraicheskoi topologii, Mir, M., 1976 | MR

[6] Kartan A., Eilenberg S., Gomologicheskaya algebra, IL, M., 1960

[7] Kuzminov V. I., “O proizvodnykh funktorakh funktora proektivnogo predela”, Sib. mat. zhurn., 8:2 (1967), 333–345 | MR

[8] Massi U., Teoriya gomologii i kogomologii, Mir, M., 1981 | MR

[9] Milnor Dzh., Stashef Dzh., Kharakteristicheskie klassy, Mir, M., 1979 | MR | Zbl

[10] Rudyak Yu. B., “Ob izomorfizme Toma–Dolda dlya neorientiruemykh rassloenii”, DAN SSSR, 255:6 (1980), 1323–1325 | MR | Zbl

[11] Svittser R M., Algebraicheskaya topologiya – gomotopii i gomologii, Nauka, M., 1985 | MR

[12] Sklyarenko E. G., “K teorii obobschennykh mnogoobrazii”, Izv. AN SSSR. Ser. mat., 35:4 (1971), 831–843 | Zbl

[13] Sklyarenko E. G., “Gomologii i kogomologii obschikh prostranstv”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 50, VINITI, M., 1989, 129–266 | MR

[14] Sklyarenko E. G., “Gomologii i kogomologii svyazi mezhdu mnozhestvami. Gomologii i kogomologii okruzheniya zamknutogo mnozhestva”, Izv. RAN. Ser. mat., 56:5 (1992), 1040–1071 | MR | Zbl

[15] Sklyarenko E. G., “O prirode gomologicheskikh umnozhenii i dvoistvennosti”, Uspekhi mat. nauk, 49:1 (1994), 141–198 | MR | Zbl

[16] Sklyarenko E. G., “Giper(ko)gomologii dlya tochnykh sleva kovariantnykh funktorov i teoriya gomologii topologicheskikh prostranstv”, Uspekhi mat. nauk, 50:3 (1995), 109–146 | MR | Zbl

[17] Sklyarenko E. G., “O kogomologiyakh s nositelyami”, Uspekhi mat. nauk, 51:1 (1996), 167–168 | MR | Zbl

[18] Sklyarenko E. G., “O gomologicheskikh umnozheniyakh”, Izv. RAN. Ser. mat., 61:1 (1997), 57–176 | MR

[19] Spaner E., Algebraicheskaya topologiya, Mir, M., 1971 | MR

[20] Sternberg S., Lektsii po differentsialnoi geometrii, Mir, M., 1970 | MR | Zbl

[21] Kharlap A. E., “Lokalnye gomologii i kogomologii, gomologicheskaya razmernost i obobschennye mnogoobraziya”, Mat. sbornik, 96:3 (1975), 347–373 | MR | Zbl

[22] Khyuzmoller D., Rassloennye prostranstva, Mir, M., 1970

[23] Bredon G. E., Sheaf theory, Second edition, Springer, 1997 ; Бредон Г. Э., Теория пучков, Наука, М., 1988 | MR | MR | Zbl

[24] Cartan H., Cohomologie des groups, suite spectral, faisceaux, Seminaire, École Normal Sup., 1950–1951

[25] Jensen C. U., Les foncteours dérivés de $\varprojlim$ et leurs applications en théorie des modules, Lect. Notes Math., 254, Springer, Berlin, New York, 1972 | MR | Zbl