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Configuration spaces and Vassiliev classes in any dimension. (English) Zbl 1029.57009

Let \(\text{Imb}(S^{1},\mathbb{R}^{n})\) be the space of smooth imbeddings of the circle \(S^{1}\) into the \(n\)-dimensional Euclidean space \(\mathbb{R}^{n}\). In the paper under review the authors study the de Rham cohomology of \(\text{Imb}(S^{1},\mathbb{R}^{n})\) by using graph cohomology.
A decorated graph consists of an oriented circle and oriented edges with labeling of vertices and edges, under certain conditions depending on the parity of \(n\). Let \(\mathcal{D}_{[n]}^{k,m}\) be the vector space over \(\mathbb{R}\) generated by decorated graphs with order \(k\) and degree \(m\) modulo some relations, where the order is minus the Euler characteristic and \(m\) measures how the graph differs from trivalent ones, and \([n]\) denotes the parity of \(n\). A coboundary operator \(\delta_{[n]}\colon\mathcal{D}_{[n]}^{k,m}\to\mathcal{D}_{[n]}^{k,m+1}\) is also defined and we have the cohomology group \(H^{k,m}(\mathcal{D}_{[n]})\).
Consider the map \(\phi_{i,j}\colon C_{q}^{0}\to S^{n-1}\) sending \((x_{1},x_{2},\dots,x_{q})\) to \(\left(x_{i}-x_{j}\right)/\left|x_{i}-x_{j}\right|\), where \(C_{q}^{0}\) is the subset of \(\mathbb{R}^{q}\) in which no pair of entries are equal. Then \(\phi_{i,j}\) induces a map from \(H^{k,m}(\mathcal{D}_{[n]})\) to the de Rham cohomology \(H^{(n-3)k+m}(\text{Imb}(S^{1},\mathbb{R}^{n}))\). The authors prove that this map is injective for any \(k\) if \(m=0\). From the combinatorial structure of \(\mathcal{D}_{[n]}^{k,m}\) they also prove that for \(n>3\) and for any positive integer \(k_{0}\) there are non-trivial classes in \(H^{k}\left(\text{Imb}(S^{1},\mathbb{R}^{n})\right)\) for some \(k>k_{0}\).
By using Chen’s iterated integral it is also shown that the inclusion map from \(\text{Imb}(S^{1},\mathbb{R}^{n})\) to the space of immersions \(\text{Imm}(S^{1},\mathbb{R}^{n})\) induces the zero map in cohomology if \(n\) is odd, and nontrivial if \(n\) is even. They also study imbeddings with framings.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
55R80 Discriminantal varieties and configuration spaces in algebraic topology
58A12 de Rham theory in global analysis
58D10 Spaces of embeddings and immersions
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References:

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