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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Symmetries of homotopy complex projective three spaces


Author: Mark Hughes
Journal: Trans. Amer. Math. Soc. 337 (1993), 291-304
MSC: Primary 57R55; Secondary 57R65, 57S17
DOI: https://doi.org/10.1090/S0002-9947-1993-1164199-5
MathSciNet review: 1164199
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Abstract: We study symmetry properties of six-dimensional, smooth, closed manifolds which are homotopy equivalent to $ {\mathbf{C}}{P^3}$. There are infinitely differentiably distinct such manifolds. It is known that if $ m$ is an odd prime, infinitely many homotopy $ {\mathbf{C}}{P^3}$'s admit $ {{\mathbf{Z}}_m}$-actions whereas only the standard $ {\mathbf{C}}{P^3}$ admits an action of the group $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$. We study the intermediate case of $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions and show that infinitely many homotopy $ {\mathbf{C}}{P^3}$'s do admit $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions for a fixed prime $ m$. The major tool involved is equivariant surgery theory. Using a transversality argument, we construct normal maps for which the relevant surgery obstructions vanish allowing the construction of $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions on homotopy $ {\mathbf{C}}{P^3}$'s which are $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-homotopy equivalent to a specially chosen linear action on $ {\mathbf{C}}{P^3}$. A key idea is to exploit an extra bit of symmetry which is built into our set-up in a way that forces the signature obstruction to vanish. By varying the parameters of our construction and calculating Pontryagin classes, we may construct actions on infinitely many differentiably distinct homotopy $ {\mathbf{C}}{P^3}$'s as claimed.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1164199-5
Keywords: Homotopy complex projective space, smooth group action, equivariant surgery theory, equivariant transversality, surgery obstruction, $ G$-signature, Pontryagin class
Article copyright: © Copyright 1993 American Mathematical Society

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