Surgery structure set

Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences ${\displaystyle f_{i}:M_{i}\to X}$ from closed manifolds ${\displaystyle M_{i}}$ of dimension ${\displaystyle n}$ to ${\displaystyle X}$ (${\displaystyle i=0,1}$) equivalent if there exists a cobordism ${\displaystyle {\mathcal {}}(W;M_{0},M_{1})}$ together with a map ${\displaystyle (F;f_{0},f_{1}):(W;M_{0},M_{1})\to (X\times [0,1];X\times \{0\},X\times \{1\})}$ such that ${\displaystyle F}$, ${\displaystyle f_{0}}$ and ${\displaystyle f_{1}}$ are homotopy equivalences. The structure set ${\displaystyle {\mathcal {S}}^{h}(X)}$ is the set of equivalence classes of homotopy equivalences ${\displaystyle f:M\to X}$ from closed manifolds of dimension n to X. This set has a preferred base point: ${\displaystyle id:X\to X}$.

There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F, ${\displaystyle f_{0}}$ and ${\displaystyle f_{1}}$ to be simple homotopy equivalences then we obtain the simple structure set ${\displaystyle {\mathcal {S}}^{s}(X)}$.

Notice that ${\displaystyle (W;M_{0},M_{1})}$ in the definition of ${\displaystyle {\mathcal {S}}^{h}(X)}$ resp. ${\displaystyle {\mathcal {S}}^{s}(X)}$ is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set ${\displaystyle {\mathcal {S}}^{s}(X)}$, provided that n>4: The simple structure set ${\displaystyle {\mathcal {S}}^{s}(X)}$ is the set of equivalence classes of homotopy equivalences ${\displaystyle f:M\to X}$ from closed manifolds ${\displaystyle M}$ of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences ${\displaystyle f_{i}:M_{i}\to X}$ (i=0,1) are equivalent if there exists a diffeomorphism (or PL-homeomorphism or homeomorphism) ${\displaystyle g:M_{0}\to M_{1}}$ such that ${\displaystyle f_{1}\circ g}$ is homotopic to ${\displaystyle f_{0}}$.

As long as we are dealing with differential manifolds, there is in general no canonical group structure on ${\displaystyle {\mathcal {S}}^{s}(X)}$. If we deal with topological manifolds, it is possible to endow ${\displaystyle {\mathcal {S}}^{s}(X)}$ with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).

Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence ${\displaystyle \phi :M\to X}$ whose equivalence class is the base point in ${\displaystyle {\mathcal {S}}^{s}(X)}$. Some care is necessary because it may be possible that a given simple homotopy equivalence ${\displaystyle \phi :M\to X}$ is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on ${\displaystyle {\mathcal {S}}^{s}(X)}$.

The basic tool to compute the simple structure set is the surgery exact sequence.

Topological Spheres: The generalized Poincaré conjecture in the topological category says that ${\displaystyle {\mathcal {S}}^{s}(S^{n})}$ only consists of the base point. This conjecture was proved by Smale (n > 4), Freedman (n = 4) and Perelman (n = 3).

Exotic Spheres: The classification of exotic spheres by Kervaire and Milnor gives ${\displaystyle {\mathcal {S}}^{s}(S^{n})=\theta _{n}=\pi _{n}(PL/O)}$ for n > 4 (smooth category).

• Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag, MR 0358813
• Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, MR 2061749
• Wall, C. T. C. (1999), Surgery on compact manifolds, Mathematical Surveys and Monographs, vol. 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388
• Ranicki, Andrew (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics 102, CUP, ISBN 0-521-42024-5, MR 1211640

• Andrew Ranicki's homepage
• Shmuel Weinberger's homepage