Recurrence relations and asymptotics for four-manifold invariants

Authors:
P. B. Kronheimer and T. S. Mrowka

Journal:
Bull. Amer. Math. Soc. **30** (1994), 215-221

MSC (2000):
Primary 57R55; Secondary 14J27, 14J29, 57N13, 57R40, 57R57

DOI:
https://doi.org/10.1090/S0273-0979-1994-00492-6

MathSciNet review:
1246469

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The polynomial invariants for a large class of smooth 4-manifolds are shown to satisfy universal relations. The relations reflect the possible genera of embedded surfaces in the 4-manifold and lead to a structure theorem for the polynomials. As an application, one can read off a lower bound for the genera of embedded surfaces from the asymptotics of for large *d*. The relations are proved using moduli spaces of singular instantons.

**[1]**S. Bauer,*Diffeomorphism types of elliptic surfaces with*, Warwick preprint, 1992.**[2]**P. J. Braam and S. K. Donaldson,*Fukaya-Floer homology and gluing formulae for polynomial invariants*, preprint. MR**1362830 (97i:57035)****[3]**S. K. Donaldson,*Polynomial invariants for smooth four-manifolds*, Topology**29**(1990), 257-315. MR**1066174 (92a:57035)****[4]**S. K. Donaldson and P. B. Kronheimer,*The geometry of four-manifolds*, Oxford Univ. Press, Oxford, 1990. MR**1079726 (92a:57036)****[5]**R. Friedman and J. W. Morgan,*Complex versus differentiable classification of algebraic surfaces*, Topology Appl.**32**(1989), 135-139. MR**1007985 (90f:57039)****[6]**-,*Smooth four-manifolds and complex surfaces*, Springer-Verlag, New York (to appear). MR**1288304 (95m:57046)****[7]**P. B. Kronheimer,*The genus-minimizing property of algebraic curves*, Bull. Amer. Math. Soc. (N.S.)**29**(1993), 63-9. MR**1193539 (94b:57037)****[8]**-,*An obstruction to removing intersection points in immersed surfaces*, Topology (to appear). MR**1432428 (97j:57039)****[9]**P. B. Kronheimer and T. S. Mrowka,*Gauge theory for embedded surfaces*I, Topology**32**(1993), 773-826. MR**1241873 (94k:57048)****[10]**-,*Gauge theory for embedded surfaces*II, Topology (to appear). MR**1308489 (96b:57038)****[11]**J. W. Morgan and T. S. Mrowka,*A note on Donaldson's polynomial invariants*, Internat. Math. Res. Notices**10**(1992), 223-230. MR**1191573 (93m:57032)****[12]**-,*On the diffeomorphism classification of regular elliptic surfaces*, Internat. Math. Res. Notices (to appear). MR**1224116 (94g:57035)****[13]**J. W. Morgan and K. G. O'Grady,*The smooth classification of fake K*3'*s and similar surfaces*, preprint, 1992.**[14]**W. P. Thurston,*A norm for the homology of*3-*manifolds*, Mem. Amer. Math. Soc., vol. 59, Amer. Math. Soc., Providence, RI, 1986, pp. 99-130. MR**823443 (88h:57014)**

Retrieve articles in *Bulletin of the American Mathematical Society*
with MSC (2000):
57R55,
14J27,
14J29,
57N13,
57R40,
57R57

Retrieve articles in all journals with MSC (2000): 57R55, 14J27, 14J29, 57N13, 57R40, 57R57

Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1994-00492-6

Article copyright:
© Copyright 1994
American Mathematical Society