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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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