Proposition

Lemma 7 ([Local counting]). Let M be a closed smooth n -manifold, possibly disconnected, and let q:M→Y be continuous, with Y Hausdorff. Suppose a point y∈Y has a neighborhood homeomorphic to an open subset of ℝ^n , and that q is smooth in …

Lemma 7 ([Local counting]). Let M be a closed smooth n -manifold, possibly disconnected, and let q:M→Y be continuous, with Y Hausdorff. Suppose a point y∈Y has a neighborhood homeomorphic to an open subset of ℝn , and that q is smooth in the corresponding coordinates near q-1 (y) , with invertible derivative at each point of that fiber. Then the fiber is finite, and the image of the total fundamental class of M in Hn(Y,Yy)≅𝔽2 is |q-1 (y)| modulo two. In particular, if Y is a connected closed n -manifold, this parity is the mod-two degree of q .

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Proposition 6 ([The number of labels]). If k≥2 , every smooth admissible map…Lemma 5 ([The odd sphere map]). The map F:Sym(V)→ℝ^m is continuous, odd, and…Proposition
The inverse function theorem makes every point of the fiber isolated. The fiber is closed in compact M , hence finite. By excision, H_n(M,M q^ -1 (y)) ≅_ x∈q^ -1 (y) H_n(M,Mx). The fundamental class has coordinate one in each summand. A local diffeomorphism sends each local generator to the generator at y . Naturality of relative homology now gives the assertion. This argument uses no smoothness away from the selected fiber.
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