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Homology of jet groups. (English) Zbl 0848.57036

The \(n\)th jet group is defined as \(J_n= \{rxa_2 x^2\dots a_n x^n\mid r\), \(a_i\in \mathbb{R}\), \(r>0\}\) together with the group operation of composition followed by truncation. The limit group \(J_\infty =\varprojlim J_n\) is the group of formal invertible series at 0 and is related to the group \(G_0^\omega\) of convergent invertible series at 0 on \(\mathbb{R}\). The second homology group of \(G_0^\omega\) is a piece of the classification of cobordism classes of real analytic \(\Gamma\)-structures on surfaces and so motivates this calculation. Let \(\mathbb{R}^+\) denote the group of positive real numbers with multiplication. There is a split exact sequence of groups \(1\to J_n'\to J_n @> D >> \mathbb{R}^+\to 1\) with \(D: J_n\to \mathbb{R}^+\) given by \(D(f)= {{df} \over {dx}} (0)\) and splitting \(\sigma: \mathbb{R}^+\to J_n\) given by \(\sigma (r)= rx\). It follows that \(D_*: H_k (J_n)\to H_k (\mathbb{R}^+)\) is an epimorphism and the authors conjectured that \(D_*\) is an isomorphism for all \(k\geq 0\).
The main theorem of the paper is a proof that \(D_*: H_2 (J_n) \to H_2 (\mathbb{R}^+)\) is an isomorphism. The tool for making the calculation is a spectral sequence associated to a short exact sequence of groups \(0\to A\to G\to Q\to 1\) with \(A\) abelian and normal in \(G\). The group \(G\) acts on \(A\) by conjugation and hence acts on the homology of \(A\). The \(E^1\)-term of the spectral sequence is given by \[ E^1_{p,q} \cong \bigoplus_{Q^p} H_q (A) \] and it converges to \(H_{p+q} (G)\). Furthermore, \(d^1\) can be explicitly given in terms of the action of \(Q\) on \(H_* (A)\).
The computation proceeds by induction and the exact sequences \hbox{\(0\to (\mathbb{R},\;) \to J_n @> p_{n-1}>> J_{n-1}\to 1\)} where \(p_{n-1}\) is projection with kernel \(\mathbb{R} \cong \{xa_n x^n\mid a_n\in \mathbb{R}\}\). To obtain the isomorphism desired, it suffices to show that \(E^2_{0,2} \cong \{0 \}\cong E^2_{1,1}\) leaving \(H_2 (J_{n-1})\) inductively isomorphic to \(H_2 (J_1) \cong H_2 (\mathbb{R}^+)\). Establishing that \(E^2_{1,1} \cong \{0\}\) involves some splendid algebra that reveals that the proof would work for any subfield of \(\mathbb{R}\) containing all roots of some positive number \(r\neq 1\). A derivation of the spectral sequence via the double complex associated to a simplicial groupoid ends the paper. The conjecture of the authors was established in greater generality by P. R. Dartnell in [ibid. 92, 109-121 (1994; Zbl 0817.57030)].

MSC:

57R99 Differential topology
20J05 Homological methods in group theory

Citations:

Zbl 0817.57030
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References:

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