On orientability and degree of Fredholm maps.(English)Zbl 1093.58003

To construct a meaningful $$\mathbb Z$$-valued degree theory for smooth maps between Banach manifolds whose differentials are Fredholm of index 0, some concept of orientability has to be introduced. Instead of requiring the manifolds to be orientable, it has been shown in [P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, J. Funct. Anal. 124, 1–39 (1994; Zbl 0802.47056); P. Benevieri and M. Furi, Ann. Sci. Math. Qué. 22, No. 2, 131–148 (1998; Zbl 1058.58502)] that one may consider the orientability of the differential instead.
In contrast to the analytic definitions of orientability devised in the papers mentioned above, here the author follows a more geometric approach: a continuous family of Fredholm operators of index 0 is called orientable if the corresponding naturally defined determinant line bundle is trivial. Under certain connectivity restrictions on the parameter space it is then shown that these notions of orientability coincide if the operator family contains an invertible operator. It should be remarked that this idea also appears in [A. Floer and H. Hofer, Math. Z. 212, 13–38 (1993; Zbl 0789.58022)].

MSC:

 58B15 Fredholm structures on infinite-dimensional manifolds 47H11 Degree theory for nonlinear operators 58C30 Fixed-point theorems on manifolds
Full Text:

References:

 [1] P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory, Ann. Sci. Math. Québec 22 (1998), 131–148. · Zbl 1058.58502 [2] ——, On the concept of orientability for Fredholm maps between real Banach manifolds, Topol. Methods Nonlinear Anal. 16 (2000), 279–306. · Zbl 1007.47026 [3] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Math. Monogr., Clarendon Press, New York, 1990. · Zbl 0820.57002 [4] K. D. Elworthy and A. J. Tromba, Differential structures and Fredholm maps in Banach manifolds, Global analysis (Berkeley, 1968), Proc. Sympos. Pure Math., 15, pp. 45–94, Amer. Math. Soc., Providence, RI, 1970. · Zbl 0206.52504 [5] D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. · Zbl 0148.43103 [6] P. M. Fitzpatrick, J. Pejsachowicz, and P. J. Rabier, Orientability of Fredholm families and topological degree for orientable non-linear Fredholm mappings, J. Funct. Anal. 124 (1994), 1–39. · Zbl 0802.47056 [7] K. A. Froyshov, An inequality for the $$h$$ -invariant in instanton Floer homology, Topology 43 (2004), 407–432. · Zbl 1045.57019 [8] J. Mawhin, Equivalence theorems for non-linear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636. · Zbl 0244.47049 [9] P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458–480. · Zbl 0052.19901 [10] G. Tian and S. Wang, Orientability and real Seiberg–Witten invariants (in preparation).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.