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A note on Alexander's duality

Published online by Cambridge University Press:  26 February 2010

B. Clarke
B. Clarke
Affiliation:
The Queen's College, Oxford.
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Extract

In [3] Pontrjagin proved the following form of the Alexander duality theorem:

Theorem A. Let K be a sub-polyhedron of the n–dimensional sphere, Sn. Let G, G* be orthogonal topological groups, G being compact. Then Hr(K; G) and Hn–r–1(Sn–K; G*) are orthogonal with the product of αεHr(K; G) and αεHn−r−1(SnK; G) determined as the linking coefficientof some cycle of class a with some cycle of class α

Type
Research Article
Information
Mathematika , Volume 3 , Issue 1 , June 1956 , pp. 40 - 46
Copyright
Copyright © University College London 1956

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References

1.Eilenberg, S. and Steenrod, N., Foundations of Algebraic Topology (Princeton, 1952).CrossRefGoogle Scholar
2.Newman, M. H. A., “Intersection Complexes I. Combinatory Theory”, Proc. Cambridge Phil. Soc., 27 (1931), 491–501.CrossRefGoogle Scholar
3.Pontrjagin, L., “The General Topological Theorem of Duality for Closed Sets”, Annals of Mathematics, 35 (1934), 904–914.CrossRefGoogle Scholar
4.Seifert, H. and Threlfall, W., Lehrbuch der Topologie (Chelsea, New York, 1947).Google Scholar
5.Spanier, E. H. and Whitehead, J. H. C., “Duality in Homotopy Theory”, Mathematika, 2 (1955), 56–80.CrossRefGoogle Scholar