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Some small aspherical spaces

Published online by Cambridge University Press:  09 April 2009

Eldon Dyer  and
A. T. Vasquez
A. T. Vasquez
Affiliation:
The Graduate SchoolCity University of New York, 33 West 42 Street New York, 10036 U. S. A.
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Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 332 - 352
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Cohen, D. E. and Lyndon, R. C., ‘Free bases for normal subgroups of free groups’, Trans. Amer. Math. Soc. 108 (1963), 526–537.CrossRefGoogle Scholar
[2]Cockcroft, W. H., ‘On two-dimensional aspherical complexes’, Proc. London Math. Soc. 4 (1954), 375–384.CrossRefGoogle Scholar
[3]Eilenberg, S. and Ganea, T., ‘On the Lusternick-Schnirelmann category of abstract groups’, Ann. Math. 65 (1957), 517–518.CrossRefGoogle Scholar
[4]Higman, G., ‘A finitely generated infinite simple group’, J. London Math. Soc. 26 (1951), 61–64.CrossRefGoogle Scholar
[5]Karrass, A., Magnus, W. and Solitar, D., ‘Elements of finite order in groups with a single defining relation’, Comm. Pure and Appl. Math. 13 (1960), 57–66.CrossRefGoogle Scholar
[6]Lyndon, R. C., ‘Cohomology theory of groups with a single defining relation’, Ann. Math. 52 (1950), 650–665.CrossRefGoogle Scholar
[7]Magnus, W., ‘Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz)’, J. reine angew. Math. 163 (1930), 141–165.CrossRefGoogle Scholar
[8]Papakyriakopoulos, C. D., ‘Attaching 2-dimensicnal cells to a complex’, Ann. Math. 78 (1963), 205–222.CrossRefGoogle Scholar
[9]Rapaport, E. S., ‘Proof of a conjecture of Papakyriakopoulos’, Ann. Math. 79 (1964), 506–513.CrossRefGoogle Scholar
[10]Stallings, J. R., ‘A finitely presented group whose 3- dimensional integral homology is not finitely presented’, Amer. J. Math. 85 (1963), 541–543.CrossRefGoogle Scholar
[11]Stallings, J. R., ‘On torsion free groups with infinitely many ends’, Ann. Math. 88 (1968), 312–334.CrossRefGoogle Scholar
[12]Steenrod, N. E., Cohomology Operations, (Princeton Univ. Press, 1962).Google Scholar
[13]Swan, R. G., ‘Groups of cohomological dimension one’, J. Algebra, 12 (1969), 585–610.CrossRefGoogle Scholar
[14]Whitehead, J. H. C., ‘On the asphericity of regions in a 3-sphere’, Fund. Math. 32 (1939), 149–166.CrossRefGoogle Scholar