##
**The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction.**
*(English)*
Zbl 1069.57019

Farrell, F. Thomas (ed.) et al., Topology of high-dimensional manifolds. Proceedings of the school on high-dimensional manifold topology, Abdus Salam ICTP, Trieste, Italy, May 21–June 8, 2001. Number 1 and 2. Trieste: The Abdus Salam International Centre for Theoretical Physics (ISBN 92-95003-12-8/pbk). ICTP Lect. Notes 9, 515-538 (2002).

The structure set of a differentiable \(n\)-manifold \(M\) is the set \(\mathbb S^{O}(M)\) of equivalence classes of pairs \((N, h)\) where \(N\) is a differentiable \(n\)-manifold and \(h : N\to M\) is a simple homotopy equivalence. Two pairs \((N, h)\) and \((N', h')\) are said to be equivalent if there exists a diffeomorphism \(f : N\to N'\) and a homotopy between \(h\) and \(h' \circ f : N\to M\). The differentiable structure set was first computed for the \(n\)-sphere (\(n\geq 5\)), with \(\mathbb S^{O}(S^n) = \Theta^n\) the Kervaire-Milnor group of exotic spheres. The Browder-Novikov-Sullivan-Wall theory for the classification of manifold structures within the simple homotopy type of a differentiable \(n\)-manifold \(M\), \(n\geq 5\), fits \(\mathbb S^{O}(M)\) into an exact sequence of pointed sets
\[
\cdots \to L_{n+1}(\mathbb Z[\pi_1(M)]) \to \mathbb S^{O}(M) \to [M, G/O] \to L_n(\mathbb Z[\pi_1(M)]) \to \cdots
\]
corresponding to the two stages of the obstruction theory for deciding whether a simple homotopy equivalence \(h : N\to M\) is homotopic to a diffeomorphism. (The primary obstruction in \([M, G/O]\) to the extension of \(h\) to a normal bordism \((f, b) : (W; M, N) \to (M\times I, M\times 0, M\times 1)\) with \(f| = id : M \to M\). The secondary obstruction \(\sigma_{*}(f, b)\in L_{n+1}(\mathbb Z[\pi_1(M)])\) to performing surgery on \((f, b)\) to make \((f, b)\) a simple homotopy equivalence, which depends on the choice of solution in the first stage). The structure set of a topological \(n\)-manifold \(M\) is the set \(\mathbb S^{TOP}(M)\) of equivalence classes of pairs \((N, h)\) where \(N\) is a topological manifold and \(h : N\to M\) is a simple homotopy equivalence. Two pairs \((N, h)\) and \((N', h')\) are said to be equivalent if there exist a homeomorphism \(f : N\to N'\) such that \(h \simeq h'\circ f : N\to M\). There is again a surgery exact sequence for \(n\geq 5\)
\[
\cdots \to L_{n+1}(\mathbb Z[\pi_1(M)]) \to \mathbb S^{TOP}(M) \to [M, G/TOP] \to L_n(\mathbb Z[\pi_1(M)]) \to \cdots
\]
corresponding to a two-stage obstruction theory for deciding whether a simple homotopy equivalence is homotopic to a homeomorphism. The work of Quinn (see F. Quinn [Bull. Am. Math. Soc. 77, 596–600 (1971; Zbl 0226.57015)]) and, subsequently, that of the author (see for example [A. A. Ranicki, Algebraic L-theory and Topological Manifolds, (Cambridge Tracts in Math. 102, Cambridge), (1992; Zbl 0767.57002)]) lead to the algebraic theory of surgery which permits to define the algebraic exact sequence of abelian groups for any topological space \(X\)
\[
\cdots \to H_n(X; \mathbb L_{\bullet}) \xrightarrow{A} L_{n}(\mathbb Z[\pi_1(X)]) \to \mathbb S_n(X) \to H_{n-1}(X; \mathbb L_{\bullet}) \to \cdots
\]
The relative groups \(\mathbb S_{*}(X)\) in this sequence are the cobordism groups of quadratic Poincaré complexes over \((\mathbb Z, X)\) which assemble to contractible quadratic Poincaré complexes over \(\mathbb Z[\pi_1(X)]\). Here a \((\mathbb Z, X)\)-category has as objects based finitely generated \(\mathbb Z\)-modules with an \(X\)-local structure. The topological surgery exact sequence of a topological \(n\)-manifold \(M\), \(n\geq 5\), was shown to be in bijective correspondence with the corresponding portion of the algebraic surgery sequence, including an explicit bijection
\[
s : \mathbb S^{TOP}(M) \to \mathbb S_{n+1}(M).
\]
The structure invariant \(s(h)\in \mathbb S_{n+1}(M)\) of a simple homotopy equivalence \(h : N\to M\) of topological \(n\)-manifolds (which is the cobordism class of a quadratic Poincaré \(n\)-complex in \((\mathbb Z, M)\) with contractible assembly over \(\mathbb Z[\pi_1(M)]\)) measures the homotopy invariant part of the failure of the point inverses \(h^{-1}(x)\) to be acyclic, for \(x\in M\). The structure invariant is such that \(s(h) = 0\) in \(\mathbb S_{n+1}(M)\) if (and for \(n \geq 5\) only if) \(h\) is homotopic to a homeomorphism (see Section 4.4 of the paper for more details). The total surgery obstruction \(s(X) \in \mathbb S_n(X)\) of a geometric Poincaré \(n\)-complex \(X\) is the cobordism class of the \(\mathbb Z[\pi_1(X)]\)-contractible quadratic Poincaré \((n-1)\)-complex \((C, \psi)\) in \(\mathbb A(\mathbb Z, X)\) with \(C = C([X] \cap \, - \, : C(X)^{n - *} \to C(X')_{* + 1})\), using the dual cells in the barycentric subdivision \(X'\) of \(X\) to regard the simplicial chain complex \(C(X')\) as a chain complex in \(\mathbb A(\mathbb Z, X)\). The total surgery obstruction \(s(X)\) measures the homotopy invariant part of the failure of the link of each simplex in \(X\) to be a homology sphere. A fundamental result says that \(s(X)\) is trivial in \(\mathbb S_n(X)\) if (and for \(n\geq 5\) only if) \(X\) is homotopy equivalent to a topological \(n\)-manifold. For a proof see Section 4.2. The quadratic \(L\)-group \(L_n(\mathbb A(\mathbb Z, X))\) is the cobordism group of a quadratic Poincaré \(n\)-complex \((C, \psi)\) in \(\mathbb A(\mathbb Z, X)\). The functor \(X\to L_{*}(\mathbb A(\mathbb Z, X))\) is the generalized homology theory with \(\mathbb L_{\bullet}(\mathbb Z)\)-coefficients
\[
L_{*}(\mathbb A(\mathbb Z, X)) = H_{*}(X; \mathbb L_{\bullet}(\mathbb Z)).
\]
The coefficient spectrum \(\overline{\mathbb L}_{\bullet} = \mathbb L_{\bullet}(\mathbb Z)\) is the special case \(R = \mathbb Z\) of a general construction. For any ring with involution \(R\) there is a \(0\)-connective spectrum \(\mathbb L_{\bullet}(R)\) such that \(\pi_{*}(\overline{\mathbb L}_{\bullet}(R)) = L_{*}(R)\), which may be constructed using quadratic Poincaré \(n\)-ads over \(R\). The assembly functor \(A : \mathbb A(\mathbb Z, X)\to \mathbb A(\mathbb Z[\pi_1(X)])\) induces assembly maps in the quadratic \(L\)-groups which fit into the \(4\)-periodic algebraic surgery exact sequence
\[
\cdots \to H_n(X; \overline{\mathbb L}_{\bullet}) \xrightarrow{A} L_{n}(\mathbb Z[\pi_1(X)]) \to \bar{\mathbb S}_n(X) \to H_{n-1}(X; \overline{\mathbb L}_{\bullet}) \to \cdots
\]
where the \(4\)-periodic algebraic structure set \(\bar{\mathbb S}_n(X)\) is the cobordism group of quadratic Poincaré \((n-1)\)-complexes \((C, \psi)\) in \(\mathbb A(\mathbb Z, X)\) such that the assembly \(A(C)\) is a simple contractible based finitely generated free \(\mathbb Z[\pi_1(X)]\)-module chain complex. See Section 4.5 for the geometric interpretation. The \(\mathbb S\)- and \(\bar{\mathbb S}\)-groups are related by an exact sequence
\[
0 \to \mathbb S_{n + 1}(X) \to \bar{\mathbb S}_{n + 1}(X) \to H_{n}(X; L_{0}(\mathbb Z)) \to \mathbb S_n(X) \to \bar{\mathbb S}_n(X) \to 0 \,.
\]
A Poincaré \(n\)-complex \(X\) has a \(4\)-periodic total surgery obstruction \(\bar{s}(X) \in \bar{\mathbb S}_n(X)\). A fundamental result of Bryant, Ferry, Mio and Weinberger (see J. Bryant, S. Ferry, W. Mio and S. Weinberger, [Ann. Math. 143, 435–467 (1996; Zbl 0867.57016)]) states that \(\bar{s}(X)\) is trivial if (and for \(n\geq 6\) only if) \(X\) is simple homotopy equivalent to a compact ANR homology manifold.

For the entire collection see [Zbl 0996.00038].

For the entire collection see [Zbl 0996.00038].

Reviewer: Fulvia Spaggiari (Modena)