Recent zbMATH articles in MSC 55R40https://zbmath.org/atom/cc/55R402024-03-13T18:33:02.981707ZWerkzeugAbelian cycles in the homology of the Torelli grouphttps://zbmath.org/1528.550112024-03-13T18:33:02.981707Z"Lindell, Erik"https://zbmath.org/authors/?q=ai:lindell.erikLet \(S_{g,r}^{s}\) be a connected, orientable surface, of genus \(g\), \(r\) marked points and \(s\) boundary components. Let \(\Gamma_{g,r}^{s}\) be its mapping class group and let \(H=H_{1}(S_{g},\mathbb{Q})\) and \(H_{\mathbb{Z}}=H_{1}(S_{g},\mathbb{Z})\). The natural action of \(\Gamma_{g,r}^{s}\) on \(S_{g}\) preserves the symplectic form and gives a homomorphism \(\Gamma_{g,r}^{s}\to Sp(H)\). The Torelli group of \(S_{g,r}^{s}\), denoted by \(\mathcal{I}_{g,r}^{s}\), is the kernel of this homomorphism. Since the image of the above homomorphism is \(Sp(H_{\mathbb{Z}})\), the natural action of this group on \(H_{n}(\mathcal{I}_{g,r}^{s})\) transforms this into an \(Sp(H_{\mathbb{Z}})\)-representation for all \(n\geq 0\). On the other hand D. Johnson constructed a homomorphism \(\tau: H_{1}(\mathcal{I}_{g,r}^{s})\to \bigwedge^{3}H_{\mathbb{Z}}\) that naturally extends to homomorphisms, for \(n\geq1\),
\[
\psi_{n}:=(\tau_{J})_{*}: H_{n}(\mathcal{I}_{g,r}^{s})\to \bigwedge^{n}(\bigwedge^{3}H).
\]
The author uses these \(\psi_{n}\) to explore properties of \(H_{n}(\mathcal{I}_{g,r}^{s})\) such as finite dimensionality, or when they are algebraic \(S_{p}(H_{\mathbb{Z}})\)-representations, and considers the injectivity of \(\psi_{n}\).
The first result is that the image \(Im(\psi_{n})\), of \(\psi_{n}\), contains all irreducible subrepresentations of weight \(3n\) for \(n\geq 1\) and \(g\geq 3n\). The second result is that \(Im(\psi_{n})\) contains many more subrepresentation defined by certain partitions in a stable range, that is for \(g\geq n+2k+l\) for certain \(k\) and \(l\). The author also proves that for \(n\geq 2\) and \(g>>0\) one has that \(im(\psi_{n})\) is a proper subgroup of \(H_{n}(\mathcal{I}_{g,r}^{s})\). The main tool is to construct certain \textit{abelian cycles} in \(H_{n}(\mathcal{I}_{g,r}^{s})\) to obtain elements in these groups and detect elements in \(Im(\psi_{n})\).
Reviewer: Daniel Juan Pineda (Michoacán)