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Whitney towers and the Kontsevich integral. (English) Zbl 1090.57013

Gordon, Cameron (ed.) et al., Proceedings of the Casson Fest. Based on the 28th University of Arkansas spring lecture series in the mathematical sciences, Fayetteville, AR, USA, April 10–12, 2003 and the conference on the topology of manifolds of dimensions 3 and 4, Austin, TX, USA, May 19–21, 2003. Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 101-134 (2004).
The question whether a space \(X\) can be immersed or embedded into a space \(Y\) necessarily imposes the condition \(\dim X \leq \dim Y\). So we may ask whether a certain kind of map \(f:X \to Y\) can be somehow, within a certain class of maps, transformed into an immersed or embedded copy of \(X\), or whether \(f(X)\) under some condition can be changed into a copy of \(X\). For given \(X\) and \(Y\) one must expect that some maps can be and some cannot be transformed into an immersion or an embedding, respectively, so there may exist obstructions to such procedures. Finding such obstructions for a given \(X\) and \(Y\) is not an easy task and the question has been around since the time of E. R. van Kampen [Abh. Math. Semin. Hamb. Univ. 9, 72–78, 152–153 (1932; Zbl 0005.02604)], H. Whitney [Ann. Math. 45, 220–246 (1944; Zbl 0063.08237)], W. T. Wu and others, and followed the development of topology in general.
Here the authors continue to develop an obstruction theory for embeddings of 2-spheres into 4-manifolds in terms of Whitney towers which they started in their article [Algebr. Geom. Topol. 1, 1–29 (2001; Zbl 0964.57022)]. These Whitney towers generalize those by T. Cochran, K. E. Orr and P. Teichner in [Ann. Math. 157, 433–519 (2003; Zbl 1044.57001)]. The work is in smooth setting. So if \(A_1, A_2, \dots ,A_m\) are generic immersions of 2-spheres into a 4-manifold \(X\), then the authors propose obstructions for changing the \(A_i\), in their regular homotopy class, to immersions with disjoint images. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees (which are basically the spines of Whitney towers), and relations coming from the 3-dimensional theory of finite type invariants.
First it is proved that the vanishing of the intersection tree invariant \(\tau_n(\mathcal W)\) for a Whitney tower \(\mathcal W\) of order \(n\) enables one to built a Whitney tower of order \(n+1\). Then, if the 2-spheres \(A_1, A_2,\dots,A_{n+2}\) admit a so-called non-repeated Whitney tower \(\mathcal W\) of order \(n\) such that non-repeated intersection tree invariant \(\lambda(\mathcal W)\) vanishes, then the homotopy classes (rel. boundary) of \(A_i\) can be represented by immersions with disjoint images. Finally, if \(L\subset S^3\) is a link, then there are unique (rel. boundary) immersions \(A_i:D^2\to D^4\) extending \(L\). This way the authors obtain link invariants in terms of Whitney towers. The result says, if \(L\) bounds a Whitney tower \(\mathcal W\) of order \(n\) in \(D^4\), then \(\tau_n(\mathcal W)=K_n(L)\), where \(K_n(L) = Z^t(L) - 1\) and \(Z^t(L)\) is the reduced Kontsevich integral of \(L\).
For the entire collection see [Zbl 1066.57002].

MSC:

57M99 General low-dimensional topology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R40 Embeddings in differential topology
57N35 Embeddings and immersions in topological manifolds
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