Images of finite maps with one-dimensional double point set. (English) Zbl 0943.57012
The Goryunov spectral sequence [V. Goryunov and D. Mond, Compos. Math. 89, No. 1, 45-80 (1993; Zbl 0839.32017)] is used to calculate the homology of the image of finite-to-one proper maps. In particular a formula for the Euler characteristic of the image of such a map is given in terms of the Euler characteristics of the multiple point spaces of the map. The \(k\)th multiple point space \({\mathcal D}^k(f)\) of a map \(f:X\to Y\) is the closure of the set
\[
\{(x_1,\dots, x_k)\in X^k\mid f(x_1)= \dots= f(x_k),\;x_i\neq x_j\}.
\]
As a corollary the Izumiya-Marar formula [S. Izumiya and W. L. Marar, J. Geom. 52, No. 1-2, 108-119 (1995; Zbl 0840.57010)] for the Euler characteristic of the image \(f(N)\) of a closed surface in a 3-manifold is obtained.
Reviewer: A.Dranishnikov (Gainesville)
MSC:
| 57N65 | Algebraic topology of manifolds |
| 58C99 | Calculus on manifolds; nonlinear operators |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
Keywords:
alternating homology; singularities of a map; Goryunov spectral sequence; Euler characteristic; Izumiya-Marar formulaReferences:
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