No Access

DEFINING AN SU(3)-CASSON/U(2)-SEIBERG–WITTEN INTEGER INVARIANT FOR INTEGRAL HOMOLOGY 3-SPHERES

YUHAN LIM

Department of Mathematics, University of California, Santa Cruz, CA 95064, USA

https://doi.org/10.1142/S0219199707002447Cited by:0 (Source: Crossref)

Abstract

An open question is the possibility of defining an integer valued SU(3)-Casson invariant for integral homology 3-spheres which involves counting the irreducible portion of the non-degenerate (perturbed) moduli space of flat SU(3)-connections plus counter-terms associated to only the non-degenerate (perturbed) reducible portion of the moduli space. The obstruction to this is the non-trivial spectral flow of a family of twisted signature operators in 3-dimensions. The parallel U(2)-Seiberg–Witten theory also has a similiar obstruction but arising from the non-trivial spectral flow of a family of twisted Dirac operators. By taking the SU(3)-flat and U(2)-Seiberg–Witten equations simultaneously the obstructions can be made to cancel and an integer invariant is obtained.

Keywords:

AMSC: 57R57

References

M. F. Atiyah, N. Hitchen and I. M. Singer, Proc. R. Soc. London A 362, 425 (1978). Crossref, Web of Science, Google Scholar
S. Akbulut and J. McCarthy , Casson's Invariant for Oriented Homology 3-Spheres — An exposition , Princeton Math. Notes 36 ( Princeton University Press , 1990 ) . Crossref, Google Scholar
M. F. Atiyah, V. K. Patodi and I. M. Singer, Math. Proc. Camb. Phil. Soc. 77, 43 (1975). Crossref, Web of Science, Google Scholar
M. F. Atiyah, V. K. Patodi and I. M. Singer, Math. Proc. Camb. Phil. Soc. 78, 405 (1975). Crossref, Web of Science, Google Scholar
M. F. Atiyah, V. K. Patodi and I. M. Singer, Math. Proc. Camb. Phil. Soc. 79, 71 (1976). Crossref, Web of Science, Google Scholar
H. Boden and C. Herald, J. Diff. Geom. 50(1), 147 (1998). Crossref, Web of Science, Google Scholar
H. Boden, C. Herald and P. Kirk, Math. Res. Lett. 8(5–6), 589 (2001). Crossref, Web of Science, Google Scholar
S. E. Cappell, R. Lee and E. Y. Miller, Bull. Amer. Math. Soc. (N.S.) 22(2), 269 (1990), DOI: 10.1090/S0273-0979-1990-15885-9. Crossref, Web of Science, Google Scholar
S. E. Cappell, R. Lee and E. Y. Miller, Comment. Math. Helv. 77(3), 491 (2002), DOI: 10.1007/s00014-002-8349-8. Crossref, Web of Science, Google Scholar
A. Floer, Comm. Math. Phys. 118, 215 (1989), DOI: 10.1007/BF01218578. Crossref, Web of Science, Google Scholar
D. Freed and K. K. Uhlenbeck , Instantons and Four-Manifolds , 2nd edn. , MSRI Publications 1 ( Springer-Verlag , 1984 ) . Crossref, Google Scholar
T. Kato , Perturbation Theory for Linear Operators , 2nd edn. ( Springer-Verlag , 1980 ) . Google Scholar
H. B. Lawson and M.-L. Michelsohn , Spin Geometry ( Princeton University Press , 1989 ) . Google Scholar
Y. Lim, Pac. J. Math. 195(1), 179 (2000). Crossref, Web of Science, Google Scholar
Y. Lim, Math. Res. Lett. 6(5–6), 631 (1999). Crossref, Web of Science, Google Scholar
Y. Lim, Geom. Topol. 7, 965 (2003), DOI: 10.2140/gt.2003.7.965. Crossref, Web of Science, Google Scholar
C. H. Taubes, J. Diff. Geom. 31, 547 (1990). Crossref, Web of Science, Google Scholar