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A streamlined proof of the convergence of the Taylor tower for embeddings in \({\mathbb R}^n\). (English) Zbl 1417.57026

Let \(M\) be a smooth manifold of dimension \(m\), and let \(\text{Emb}(M,{\mathbb{R}}^n) \) denote the space of smooth embeddings of \(M\) into \({\mathbb{R}}^n.\) In [Geom. Topol. 3, 67–101 (1999; Zbl 0927.57027)], M. Weiss constructed a tower of approximations of the space of embeddings \begin{align*} \text{Emb}(M,{\mathbb{R}}^n) \to & \cdots \to T_{k+1}\text{Emb}(M,{\mathbb{R}}^n)\to T_{k}\text{Emb}(M,{\mathbb{R}}^n) \to\\ & \cdots \to T_{1}\text{Emb}(M,{\mathbb{R}}^n)\simeq \text{Imm}(M,{\mathbb{R}}^n). \end{align*} Informally speaking, \(T_{k}\text{Emb}(M,{\mathbb{R}}^n)\) is the “\(k\)-th Taylor polynomial” of the functor \(M\mapsto \text{Emb}(M,{\mathbb{R}}^n).\) This construction of Weiss has had many applications and has been an object of intense study.
Just as with ordinary Taylor series, questions of convergence are important both for theory and applications. In this paper the authors give elementary and self-contained proofs of the following two convergence results:
1. The map \( T_{k+1}\text{Emb}(M,{\mathbb{R}}^n)\to T_{k}\text{Emb}(M,{\mathbb{R}}^n) \) between the stages in the Taylor tower is \((k(n-m-2)-m+1)\)-connected. It follows that if \(n>m+2\), then the Taylor tower converges strongly to its limit.
2. If \(n>2m+2\), then the Taylor tower converges to \(\text{Emb}(M,{\mathbb{R}}^n).\)
A much stronger and harder theorem of Goodwillie and Klein says that the map \[ \text{Emb}(M,{\mathbb{R}}^n)\to T_{k}\text{Emb}(M,{\mathbb{R}}^n) \] is \((k(n-m-2)-m+1)\)-connected, see [T. G. Goodwillie and J. R. Klein, J. Topol. 8, No. 3, 651–674 (2015; Zbl 1329.57029)]. This theorem implies that if \(n>m+2\), then the Taylor tower converges strongly to \(\text{Emb}(M,{\mathbb{R}}^n).\)

MSC:

57R40 Embeddings in differential topology
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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