## On the calculation of UNil.(English)Zbl 1080.57025

Let $$X$$ be a closed $$n$$–dimensional manifold, expressed as a union of codimension 0 submanifolds $$X=X_1 \cup X_{-1}$$, with $$X_0=X_1 \cap X_{-1}=\partial X_1 =\partial X_{-1}\subset X$$ a codimension 1 submanifold, and $$\pi_1(X) =\pi_1(X_1) *_{\pi_1(X_0)} \pi_1(X_{-1})$$. Given another closed $$n$$–dimensional manifold $$M$$ and a simple homotopy equivalence $$f\colon M \to X$$, the obstruction to deforming $$f$$ by an $$h$$–cobordism of domains to a homotopy equivalence of the form $$f_1 \cup f_{-1} \colon M_{1}\cup M_{-1} \to X_1 \cup X_{-1}$$, with $$f_{\pm 1}\colon (M_{\pm 1}, \partial M_{\pm 1}) \to (X_{\pm 1}, \partial X_{\pm 1})$$ homotopy equivalences of manifolds with boundary such that $$f_1| =f_{-1}| \colon \partial M_1 =\partial M_{-1} \to \partial X_1 =\partial X_{-1}$$, is an element $$s(f)$$ belonging to the unitary nilpotent $$L$$–group $$UNil_{n+1}(R; \mathcal B_1, \mathcal B_{-1})$$, where $$R=\mathbb Z[\pi_1(X_0)]$$ and $$\mathcal B_{\pm 1}=\mathbb Z[\pi_1(X_{\pm 1})/\pi_1(X_0)]$$. In the paper under review the authors provide a new description of $$UNil_n(R)=UNil_n(R;R,R)$$ for any ring with involution $$R$$, in terms of $$L$$–groups, which turns out to be useful for computation. Using the quadratic Poincaré cobordism formulation of the $$L$$–groups they prove the following decomposition $L_n(R[x])=L_n(R) \oplus UNil_n(R).$ Then they relate this result to the group of symmetric structures on the universal chain bundle (see M. Weiss, Proc. Lond. Math. Soc. 51, 146–192, 193–230 (1985; Zbl 0617.57019)]) to produce almost complete calculations of $$UNil_*(\mathbb Z;\mathbb Z, \mathbb Z)$$ and the Wall surgery obstruction groups $$L_*(\mathbb Z[D_{\infty}])$$ of the infinite dihedral group $$D_{\infty}=\mathbb Z_2 * \mathbb Z_2$$.

### MSC:

 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 57R67 Surgery obstructions, Wall groups

### Keywords:

L-theory; UNil-groups; infinite dihedral group

Zbl 0617.57019
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### References:

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