Journal of the Mathematical Society of Japan

On the nonuniqueness of equivariant connected sums

Mikiya MASUDA and Reinhard SCHULTZ

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Abstract

In both ordinary and equivariant 3-dimensional topology there are strong uniqueness theorems for connected sum decompositions of manifolds, but in ordinary higher dimensional topology such decompositions need not be unique. This paper constructs families of manifolds with smooth group actions that are equivariantly almost diffeomorphic but have infinitely many inequivalent equivariant connected sum representations for which one summand is fixed. The examples imply the need for restrictions in any attempt to define Atiyah-Singer type invariants for odd dimensional manifolds with nonfree smooth group actions. Applications to other questions are also considered.

Article information

Source
J. Math. Soc. Japan, Volume 51, Number 2 (1999), 413-435.

Dates
First available in Project Euclid: 10 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213108025

Digital Object Identifier
doi:10.2969/jmsj/05120413

Mathematical Reviews number (MathSciNet)
MR1674757

Zentralblatt MATH identifier
0928.57036

Subjects
Primary: 57S15: Compact Lie groups of differentiable transformations 57S17: Finite transformation groups
Secondary: 57R55: Differentiable structures 57R67: Surgery obstructions, Wall groups [See also 19J25]

Keywords
Connected sum equivariant almost diffeomorphism equivariant inertia group tangential representations at fixed points Gap Hypothesis generalized Atiyah-Singer invariants semifree circle actions

Citation

MASUDA, Mikiya; SCHULTZ, Reinhard. On the nonuniqueness of equivariant connected sums. J. Math. Soc. Japan 51 (1999), no. 2, 413--435. doi:10.2969/jmsj/05120413. https://projecteuclid.org/euclid.jmsj/1213108025


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