On Zeeman's filtration in homology

E. G. Sklyarenko

Sbornik. Mathematics, Tome 77 (1994) no. 2, pp. 477-488

Résumé

For a finite complex $K$, Zeeman constructed a spectral sequence, converging to the homology of the complex, of the form $E_2^{pq}=H^q(K;\mathcal H_p)\Rightarrow H_{p-q}(K)$. Special attention was given to the corresponding filtration in the homology of $K$, essentially dependent on the cohomology: \begin{gather*} H_r(K)=F^0H_r(K)\supset F^1H_r(K)\supset\dots\supset F^qH_r(K)\supset \dots, \\ E_\infty^{pq}=F^qH_r(K)/F^{q+1}H_r(K),\qquad r=p-q, \end{gather*} where $\mathcal H_p$ is the coefficient system determined by the local homology groups $H_p^x=H_p(K,\,K\setminus x)$. The object of the present paper is to show that the Zeeman filtration, although defined in terms of the simplicial structure of the complex, is, in the end, of a general-categorical nature. Due to this fact, a more complete description of its connection with the topology of the space and with the product is obtained.

Bibliographie

[1] Petkova S. V., “Ob aksiomakh teorii gomologii”, Matem. sb., 90:4 (1973), 607–624 | MR | Zbl

[2] Petkova S. V., “O sovpadenii gomologii na gomologicheski lokalno svyaznykh prostranstvakh”, Dokl. Bolg. AN, 35:4 (1982), 427–430 | MR | Zbl

[3] Sklyarenko E. G., “Teoriya gomologii i aksioma tochnosti”, UMN, 24:5 (1969), 87–140 | MR | Zbl

[4] Sklyarenko E. G., “K teorii gomologii, assotsiirovannoi s kogomologiyami Aleksandrova–Chekha”, UMN, 34:6 (1979), 90–118 | MR | Zbl

[5] Sklyarenko E. G., “Dvoistvennost Puankare i sootnosheniya mezhdu funktorami Tor i Ext”, Matem. zametki, 28:5 (1980), 769–776 | MR | Zbl

[6] Sklyarenko E. G., “Gomologii i kogomologii obschikh prostranstv”, Itogi nauki i tekhn. Sovrem. probl. matem. Fundam. napravleniya, 50, VINITI, 1989, 129–266 | MR

[7] Sklyarenko E. G., “Obschie teorii gomologii i kogomologii. Sovremennoe sostoyanie i tipichnye primeneniya”, Itogi nauki i tekhn. Algebra, topologiya, geometriya, 27, VINITI, 1989, 125–228 | MR | Zbl

[8] Bredon G. E., Sheaf theory, McGraw–Hill, New York, 1967 ; Bredon G. E., Teoriya puchkov, Nauka, M., 1988 | MR | Zbl | MR

[9] Cartan H., Cohomologie des groups, suite spectral, faisceaux, Séminaire Paris, Ecole Norm. Super., 1950–1951

[10] Cartan H., Eilenberg S., Homological algebra, Univer. Press, Princeton, 1956 ; Kartan A., Eilenberg S., Gomologicheskaya algebra, IL, M., 1960 | MR | Zbl

[11] Godement R., Topologie algebraique et théorie des faisceaux, Hermann, Paris ; Godeman R., Algebraicheskaya topologiya i teoriya puchkov, IL, M., 1961 | MR

[12] Grothendieck A., “Sur le quelques points d'algébre homologique”, Tohoku Math. J., 9 (1957), 119–211 ; Grotendik A., O nekotorykh voprosakh gomologicheskoi algebry, IL, M., 1961 | MR | Zbl

[13] Massey W. S., Homology and cohomology theory, M. Dekker inc.; Basel, New York, 1978 ; Massi U., Teoriya gomologii i kogomologii, Mir, M., 1981 | MR | Zbl

[14] McGrory C., “Zeeman's filtrations of homology”, Trans. Amer. Math. Soc., 250 (1979), 147–166 | DOI | MR

[15] Schapira P., Theorie des Hyperfonctions, Springer, Berlin–Heidelberg–New York, 1970 ; Shapira P., Teoriya giperfunktsii, Mir, M., 1972 | MR | Zbl | MR | Zbl

[16] Swan R. G., The theory of sheaves, Math. Inst. Univ. of Oxford, 1958

[17] Whitehead G., “Generalized homology theories”, Trans. Amer. Math. Soc., 102 (1962), 227–283 | DOI | MR | Zbl

[18] Zeeman E. C., “Dihomology”, Proc. International Congress, Amsterdam, 1954

[19] Zeeman E. C., “Dihomology. III: A generalization of the Poincare duality for manifolds”, Proc. London Math. Soc., 13:49 (1963), 155–183 | DOI | MR | Zbl