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Finite differences of Rokhlin’s function. (Russian. English summary) Zbl 0754.57029
Let \(M\) be an \(8k+3\)-dimensional, smooth, closed manifold which is a boundary. The Rokhlin function on the set \(\text{Spin}(M)\) of spinor structures on \(M\) is the function \(R_ M\): \(\text{Spin}(M)\to \mathbb{Z}/16\), \(R_ M(\varphi)=\sigma(W)\bmod 16\) where \(\sigma(W)\) is the signature of the compact spinor manifold \(W\) bounded by \(M\), with the spinor structure \(\varphi\). Consider the action \(\varphi\to \varphi+h\) of \(H^ 1(M,\mathbb{Z}/2)\) on \(\text{Spin}(M)\) and define the difference functions \(\Delta R_ M(\varphi;h)=R_ M(\varphi+h)-R_ M(\varphi),\dots,\Delta^{n+1}(\varphi;h_ 0,\dots ,h_ n)=\Delta^ n R_ M(\varphi+h_ 0; h_ 1,\dots,h_ n)-\Delta^ nR_ M(\varphi;h_ 1,\dots,h_ n)\). Theorem. For every manifold \(M\) of dimension \(n=8k+3\), \(\text{deg }R_ M\leq n\). Theorem. \(\Delta^ 3R_ M\equiv 0{}\bmod 8\). The degrees of \(R_ M\) are estimated by using some invariants of generalized Pin-structures.
Reviewer: V.Oproiu (Iaşi)
MSC:
57R85 Equivariant cobordism
57N10 Topology of general \(3\)-manifolds (MSC2010)
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