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Homology of spaces of knots in any dimensions. (English) Zbl 0994.55010
The paper deals with standard compact knots, that is, smooth embeddings \(\mathbb S^1\to {\mathbb R}^n\), and long knots, that is, embeddings \({\mathbb R}^1\to{\mathbb R}^n\) coinciding with a standard linear embedding outside some compact subset in \({\mathbb R}^1\). Let \(\mathcal K\) denote the space of all smooth maps \({\mathbb S}^1\to {\mathbb R}^n\) (respectively, \({\mathbb R}^1\to {\mathbb R}^n\) with such boundary conditions). It turns out that \(\mathcal K\) is a linear (respectively, an affine) space. The discriminant \(\Sigma \subset \mathcal K\) is the set of all maps which are not smooth embeddings, that is, they have either self-intersections or singular points. Of course, the space of knots is precisely the complement \(\mathcal K\smallsetminus \Sigma\). This interesting paper represents a survey on the recent results, obtained by various authors (see the references), on the computation of the cohomology groups of the space \(\mathcal K\smallsetminus \Sigma\). It is convenient to study this cohomology by a sort of Alexander duality \(\widetilde H^i(\mathcal K\smallsetminus \Sigma)\cong \overline H_{n\infty-i-1}(\Sigma)\), where \(\overline H_{*}\) denotes the Borel-Moore homology, that is, the homology group of the one-point compactification, and \(n\infty\) represents the dimension of \(\mathcal K\). Note that the meaning of the right-hand side in the above isomorphism follows from the fact that the topology of discriminants can be studied by means of simplicial (or, more generally, conical) resolutions. These filtrations give rise to cohomological spectral sequences with the support in the second quadrant. It was proved that, if \(n\geq 4\), then the spectral sequence converges exactly to the cohomology group \(H^{*}(\mathcal K\smallsetminus \Sigma)\) of the space of knots in \(\mathbb R^n\). The question is still open for \(n=3\); however, it is noteworthy to observe that the knot invariants appear as zero-dimensional cohomology classes of the space of knots in \(\mathbb R^3\). Finally, combinatorial expressions of knot invariants and some computations of the first positive-dimensional cohomology class of the (resolved) space of long knots in \(\mathbb R^3\) complete the paper.

MSC:
55R80 Discriminantal varieties and configuration spaces in algebraic topology
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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