Sequential homology

Resumen

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity'. An exterior map is a continuous map which is 'continuous at infinity'. A strongly locally finite CW-complex X, whose skeletons are provided with the family of the complements of compact subsets, can be considered as an exterior space X̄. Associated with a compact metric space we also consider the open fundamental complex OFC(X)̄; introduced by Lefschetz. In this paper we use sequences of cycles converging to infinity to introduce 'ordinary' sequential homology and cohomology theories in the category of exterior spaces. One of the interesting differences with respect to the ordinary theories of topological spaces is that the role of a point is played by the exterior space N of natural numbers with the discrete topology and the cofinite externology. For a strongly locally finite CW-complex X, we see that the singular homology of X is isomorphic to H•seq(X̄; ⊕0∞ ℤ), the locally finite homology is isomorphic to H•seq(X̄; π0∞ ℤ) and the end homology is isomorphic to H•seq(X̄; π0∞ ℤ/⊕0∞ ℤ cohomology one has that the compact support cohomology is isomorphic to H•seq(X̄; ⊕0∞ ℤ), the singular cohomology is isomorphic to H•seq(X̄; π0∞ ℤ ) and the end cohomology is isomorphic to H•seq(X̄; π0∞ℤ). With respect to the Lefschetz fundamental complex, one has that the Čech homology of a compact metric space can be found as a subgroup of H•seq(OFC(X); ℛ)̄; the Steenrod homology is isomorphic to H•+1seq (OFC(X);̄; π0∞ ℤ/⊕0∞ ℤ) and the Čech cohomology of X is isomorphic to Hseq•(OFC(X);̄; π0∞ ℤ/⊕0∞ ℤ). Finally, one also has a Poincaré isomorphism Hseqq(M̄) ≅ Hn-qseq (M̄), where M is a triangulable, second countable, orientable, n-manifold. We remark that in both sides of the isomorphism we are using sequential theories. © 2001 Elsevier Science B.V. All rights reserved.