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topological index

Context

Index theory

Integration theory

Contents

Idea

For a given manifold XX of finite dimension there exists an embedding i:X ni : X \to \mathbb{R}^n into some Cartesian space. Using the Pontrjagin-Thom collapse map this induces a morphism in topological K-theory

i !:K(TX)K(T n). i_! : K(T X) \to K(T \mathbb{R}^n) \,.

Similarly for any point inclusion j:* nj : * \to \mathbb{R}^n there is such a morphism j !:=K(*)K(T n)j_! : \mathbb{Z} = K(*) \to K(T \mathbb{R}^n) which is an isomorphism – the Thom isomorphism.

The topological index of topological K-theory on XX is the composite

ind top:=K(TX)i !K(T n)j ! 1K(*)=. ind_{top} : \mathbb{Z} = K(T X) \stackrel{i_!}{\to} K(T \mathbb{R}^n) \stackrel{j_!^{-1}}{\to} K(*) = \mathbb{Z} \,.

One can prove that this is independent of all the occurring choices. In particular it does not depend on the specific choice of embedding of the manifold XX into to the Euclidean space. The topological index function is uniquely fixed by two properties (this is the content of the Atiyah-Singer index theorem):

  1. For XX a point we have ind t=idind_t=id.

  2. Index functions commute with the maps i !i_!.

From this one defines the topological index of an elliptic operator . The principal symbol of the operator defines a homogenous length-one chain complex of bundles on TXT X exact outside the null section. Elements of this kind are precisely cycles for the compactly supported K-theory of TXT X hence an elliptic operator DD has a topological index only depending on its principal symbol.

On the other hand, analysis associates to DD its analytical index that is

dimKer(D)dimcoKer(D) \operatorname{dim}\operatorname{Ker}(D)-\operatorname{dim}\operatorname{coKer}(D)

on one (hence all) Sobolev space that DD is defined on.

The Atiyah-Singer index theorem states that the analytical index of DD is equal to its topological index.

More on the Thom map

The story starts with an embedding i:XYi:X\to Y of compact manifolds. In this situation one can construct a homomorphism

i !:K(TX)K(TY) i_{!} : K(T X)\longrightarrow K(T Y)

between the compactly supported K-theories of their tangent bundles.

Notice here the reverse functoriality: for the base space KK is contravariant while for the total spaces of the tangent bundles it is covariant. This uses the Thom mapping: if XX is a compact manifold and VV a real vector bundle over XX there is a natural map

φ:K(X)K(V). \varphi:K(X)\longrightarrow K(V) \,.

One of the most important results of K-theory, namely Bott periodicity, can be seen as the statement of the fact that this map is an isomorphism. Now apply this construction to the normal bundle NN of XX in YY to get

φ:K(TX)K(TN) \varphi : K(T X)\longrightarrow K(T N)

and (looking at NN as a tubular neighbourhood of XX in YY) compose it with the natural map

K *:K(TN)K(TY) K_* : K(T N)\longrightarrow K(T Y)

to get i !i_!.

Now given a manifold XX, embed it in a Euclidean space n\mathbb{R}^n for some suitable nn and consider the inclusion {0} n\{0\}\to \mathbb{R}^n. This induces the (Thom isomorphism) mapping j !:=K({0})K(T n)j_!:\mathbb{Z}=K(\{0\}) \longrightarrow K(T\mathbb{R}^n).

The topological index is defined to be

ind t:=j ! 1i !. ind_t := j_!^{-1}\circ i_! \,.

Revised on August 29, 2014 05:26:51 by Urs Schreiber (89.204.138.134)