Moving homology classes to infinity. (English) Zbl 1158.55005
Let \(q: \widetilde X \to X\) be a regular cover of a finite CW space \(X\) with the group \(\mathbb Z^r\) of covering transformations, and let \(A: \widetilde X \to \mathbb R^r\) be the Abel–Jacobi map for \(q\). Since \(\mathbb R^r=(\pi_1(X)/{\text{Im}} q_*)\otimes \mathbb R\), any class \(\xi\in H^1(X;\mathbb R)\) with \(q^*\xi=0\) yields a map \(\xi_{\mathbb R}:\mathbb R^r\to \mathbb R\). A neighborhood of infinity with respect to \(\xi\) is a subset of \(\widetilde X\) that contains a set \(\{x\in \widetilde X: \xi_{\mathbb R}(A(x))>c\}\) for some \(c\in \mathbb R\). The authors characterize homology classes that can be moved into an arbitrary small neighborhood of infinity.
Reviewer: Yuli Rudyak (Gainesville)
MSC:
55N25 | Homology with local coefficients, equivariant cohomology |
55U99 | Applied homological algebra and category theory in algebraic topology |
References:
[1] | Farber M., Topol. Methods Nonlinear Anal. 19 pp 123– (2002) |
[2] | Farber M., Lusternik-Schnirelmann Category and Related Topics. Contemporary Mathematics 316 pp 95– (2002) |
[3] | DOI: 10.1016/0040-9383(79)90033-8 · Zbl 0416.57013 · doi:10.1016/0040-9383(79)90033-8 |
[4] | Novikov S., Soviet. Math. Doklady 24 pp 222– (1981) |
[5] | DOI: 10.1007/BF02564438 · Zbl 0684.58015 · doi:10.1007/BF02564438 |
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