In the mathematical surgery theory, the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension > 4 {\displaystyle >4} . The surgery structure set S ( X ) {\displaystyle {\mathcal {S}}(X)} of a compact n {\displaystyle n} -dimensional manifold X {\displaystyle X} is a pointed set which classifies n {\displaystyle n} -dimensional manifolds within the homotopy type of X {\displaystyle X} . The basic idea is that in order to calculate S ( X ) {\displaystyle {\mathcal {S}}(X)} it is enough to understand the other terms in the exact sequence, which are usually easier to determine. These are on one hand the normal invariants which form generalized cohomology groups, and hence one can use standard tools of algebraic topology to calculate them at least in principle. On the other hand, there are the L-groups which are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. A great deal is known about these groups. Another part of the sequence are the surgery obstruction maps from normal invariants to the L-groups. For these maps there are certain characteristic classes formulas, which enable to calculate them in some cases. Knowledge of these three components, that means: the normal maps, the L-groups and the surgery obstruction maps is enough to determine the structure set (at least up to extension problems). In practice one has to proceed case by case, for each manifold
X {\displaystyle {\mathcal {}}X} it is a unique task to determine the surgery exact sequence, see some examples below. Also note that there are versions of the surgery exact sequence depending on the category of manifolds we work with: smooth (DIFF), PL, or topological manifolds and whether we take Whitehead torsion into account or not (decorations s {\displaystyle s} or h {\displaystyle h} ). The original 1962 work of Browder and Novikov on the existence and uniqueness of manifolds within a simply-connected homotopy type was reformulated by Sullivan in 1966 as a surgery exact sequence. In 1970 Wall developed non-simply-connected surgery theory and the surgery exact sequence for manifolds with arbitrary fundamental group.
Definition The surgery exact sequence is defined as
⋯ → N ∂ ( X × I ) → L n + 1 ( π 1 ( X ) ) → S ( X ) → N ( X ) → L n ( π 1 ( X ) ) {\displaystyle \cdots \to {\mathcal {N}}_{\partial }(X\times I)\to L_{n+1}(\pi _{1}(X))\to {\mathcal {S}}(X)\to {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))}
where: the entries N ∂ ( X × I ) {\displaystyle {\mathcal {N}}_{\partial }(X\times I)} and N ( X ) {\displaystyle {\mathcal {N}}(X)} are the abelian groups of normal invariants, the entries
L n + 1 ( π 1 ( X ) ) {\displaystyle {\mathcal {}}L_{n+1}(\pi _{1}(X))} and
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