Surgery formulae for analytical invariants of manifolds (Q2765022)

This survey paper was presented at a conference in honor of Jim Milgram's 60th birthday. Its main subject is splitting and surgery formulae which arise for analytically defined invariants of manifolds. NEWLINENEWLINENEWLINEIn sections 2 and 3, two typical examples from the author's previous works, of the analytical tools used for and the results obtained from such formulae are demonstrated. The first is that of the exponential decay of eigenmodes with the length of a long cylinder via which two manifolds are glued together. The second has applications for the equivalence different definitions of the Casson-Walker invariant. NEWLINENEWLINENEWLINEThe remainder of the paper concentrates on the Seiberg-Witten-Casson invariant, \(\lambda_{SW}\) and gives a proof, illustrating the ideas of the first two sections, modulo a paper of \textit{A. L. Carey, M. Marcolli} and \textit{B.-L. Wang} [Exact triangles in Seiberg-Witten-Floer theory, Part I: The geometric triangle; preprint math.DG/9907065] of the Kronheimer-Mrowka conjecture of the equality of this with the usual Casson invariant. NEWLINENEWLINENEWLINEMore precisely, \textit{W. Chen} [Turk. J. Math. 21, No. 1, 61-81 (1997; Zbl 0891.57021)], \textit{Y. Lim} [Math. Res. Lett. 6, No 5-6, 631-643 (1999; Zbl 0948.57007)] and Marcolli-Wang [preprint arXiv.dg-ga/9606003] showed how the Seiberg-Witten equations on an oriented \(\mathbb Z\) homology sphere \(Y\), could be systematically deformed, dependent on some data \((g,v)\), so as to obtain a moduli space of solutions which consists of oriented points and thus has an Euler characteristic \(\chi(M^*_{SW}(Y,g,v))\). This is unfortunately not an invariant of \(Y\), that is, it depends on \((g,v)\), but can be modified to give an invariant, \(\lambda_{SW}(Y)\), by adding correction terms (expressed as spectral flows and eta invariants). The preprint of Carey-Marcolli-Wang gives the behavior under surgery of \(\chi(M^*_{SW}(Y,g,v))\), while this paper gives a proof that the correction terms behave appropriately so that the combined invariant obeys the same surgery law as does the standard Casson invariant, hence identifying the two invariants. See also \textit{M. Marcolli} and \textit{B.-L. Wang} [Geom. Dedicata 91, 45-58 (2002; Zbl 0994.57027)] for another such proof.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00054].