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.