nLab Kirby-Siebenmann invariant

Idea

The Kirby-Siebenmann invariant (or Kirby-Siebenmann class) controls the existence of PL structures on topological manifolds. The analogue concept controlling the existence of smooth structures on topological manifolds and piecewise linear (PL) manifolds are the Kervaire-Milnor groups.

Definition

Let $Top_n\coloneqq C^0(\mathbb{R}^n,\mathbb{R}^n)$ be the topological group of homeomorphisms (homeomorphism group?) and $PL_n$ the topological group of PL homeomorphisms? of Euclidean space $\mathbb{R}^n$. There is a canonical inclusion $PL_n\hookrightarrow Top_n$. Taking the cartesian product with the identity furthermore gives inclusions $Top_n\hookrightarrow Top_{n+1}$ and $PL_n\hookrightarrow PL_{n+1}$. An inductive limit yields topological groups:

$$Top \coloneqq\lim_{n\rightarrow\infty}Top_n;$$

$$PL \coloneqq\lim_{n\rightarrow\infty}PL_n.$$

There is an induced canonical inclusion $PL\hookrightarrow Top$. Their classifying spaces can be now be regarded to study the different structures: For a topological manifold $X$, its tangent bundle $TX$ is also a topological manifold, which is classified by a continuous map $X\rightarrow BTop$. Analogous for a PL manifold, there is a classifying map $X\rightarrow BPL$. The canonical inclusion $BPL\hookrightarrow BTop$ shows that every PL is a topological structure.

The quotient group $PL/Top$ only has a single non-trivial homotopy group: (Freed & Uhlenbeck 91, p. 12-13)

$$\pi_3(PL/Top) \cong\mathbb{Z}_2$$

and hence is a model for the Eilenberg-MacLane space $K(\mathbb{Z}_2,3)$. The quotient group $BPL/BTop$ now also only has a single non-trivial homotopy group since the classifying space shifts these one up:

$$\pi_4(BPL/BTop) \cong\mathbb{Z}_2.$$

and hence is a model for the Eilenberg-MacLane space $K(\mathbb{Z}_2,4)$. For any topological space $X$, the fiber bundle $BPL\hookrightarrow BTop\twoheadrightarrow BPL/BTop\simeq K(\mathbb{Z}_2,4)$ induces a short exact sequence:

$$[X,BPL] \rightarrow[X,BTop] \xrightarrow{\kappa}[X,BPL/BTop] \cong[X,K(\mathbb{Z}_2,4)] \cong H^4(X,\mathbb{Z}_2).$$

For any topological manifold $X$, the homotopy class of its classifying map $X\rightarrow BTop$ is in the middle set. A compatible PL structure exists if it results from the restriction of the homotopy class of a classifying map $X\rightarrow BPL$ from the left set, hence is in the image of the former map. Due to exactness, this is equivalent to the latter map sending it to the trivial cohomology class in the right group. Now this map is the Kirby-Siebenmann invariant (or Kirby-Siebenmann class):

$$\kappa\colon[X,BTop]\rightarrow H^4(X,\mathbb{Z}_2).$$

A topological manifold $X$ has a compatible PL structure if and only if $\kappa(X)=0$. A particular interesting case is topological 4-manifolds $X$ with $H^4(X,\mathbb{Z}_2)\cong\mathbb{Z}_2$ and therefore a binary Kirby-Siebenmann invariant.

References

Named after the original discussion in:

• Robion C. Kirby, Laurence C. Siebenmann, pages 23, 202, 252, 318 in: Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Princeton University Press (1977) [doi:10.1515/9781400881505, jstor:j.ctt1b9s024, pdf]

Further descriptions:

• Daniel Freed, Karen Uhlenbeck, Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)

See also:

• Wikipedia, Kirby-Siebenmann invariant