On 2-Cohomotopy cocycle spaces of (hence of mapping spaces into the 2-sphere from) surfaces:
Vagn Lundsgaard Hansen: On the Space of Maps of a Closed Surface into the 2-Sphere, Mathematica Scandinavica 35 (1974) 149-158 [doi:10.7146/math.scand.a-11542 , jstor:24490694, dml:166381]
Vagn Lundsgaard Hansen: The Homotopy Groups of a Space of Maps between Oriented Closed Surfaces, Bulletin on the London Mathematical Society 15 4 (1983) [doi:10.1112/blms/15.4.360]
and more generally on -Cohomotopy cocycle spaces of -manifolds:
Vagn Lundsgaard Hansen, The homotopy problem for the components in the space of maps on the -sphere, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 [doi:10.1093/qmath/25.1.313]
Vagn Lundsgaard Hansen: On spaces of maps of -manifolds into the -sphere, Trans. Amer. Math. Soc. 265 (1981) 273-281 [doi:10.1090/S0002-9947-1981-0607120-X]
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