Abstract
We study questions related to the orientability of the infinite-dimensional moduli spaces formed by solutions of elliptic equations for mappings of manifolds. The principal result states that the first Stiefel–Whitney class of such a moduli space is given by the ℤ2-spectral flow of the families of linearised operators. Under an additional compactness hypotheses, we develop elements of Morse–Bott theory and express the algebraic number of solutions of a non-homogeneous equation with a generic right-hand side in terms of the Euler characteristic of the space of solutions corresponding to the homogeneous equation. The applications of this include estimates for the number of homotopic maps with prescribed tension field and for the number of the perturbed pseudoholomorphic tori, sharpening some known results.
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Mathematics Subject Classifications (2000): 35J05, 58B15, 58E05, 58E20, 53D45
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Kokarev, G., Kuksin, S. Quasi-linear elliptic differential equations for mappings of manifolds, II. Ann Glob Anal Geom 31, 59–113 (2007). https://doi.org/10.1007/s10455-006-9034-8
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DOI: https://doi.org/10.1007/s10455-006-9034-8