Local system

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In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in a topological space X. Such a concept was introduced by Norman Steenrod in 1943.[1]

Definition[edit]

Let X be a topological space. A local system (of abelian groups/modules/...) on X is a locally constant sheaf (of abelian groups/modules...) on X. In other words, a sheaf is a local system if every point has an open neighborhood such that is a constant sheaf.

Equivalent definitions[edit]

Path-connected spaces[edit]

If X is path-connected, a local system of abelian groups has the same fibre L at every point. To give such a local system is the same as to give a homomorphism

and similarly for local systems of modules,... The map giving the local system is called the monodromy representation of .

Proof of equivalence —

Take local system and a loop at x. It's easy to show that any local system on is constant. For instance, is constant. This gives an isomorphism , i.e. between L and itself. Conversely, given a homomorphism , consider the constant sheaf on the universal cover of X. The deck-transform-invariant sections of gives a local system on X. Similarly, the deck-transform-ρ-equivariant sections give another local system on X: for a small enough open set U, it is defined as

where is the universal covering.

This shows that (for X path-connected) a local system is precisely a sheaf whose pullback to the universal cover of X is a constant sheaf.

Stronger definition on non-connected spaces[edit]

Another (stronger, nonequivalent) definition generalising 2, and working for non-connected X, is: a covariant functor

from the fundamental groupoid of to the category of modules over a commutative ring . Typically . What this is saying is that at every point we should assign a module with a representation such that these representations are compatible with change of basepoint for the fundamental group.

Examples[edit]

  • Constant sheaves. For instance, . This is a useful tool for computing cohomology since the sheaf cohomology
    is isomorphic to the singular cohomology of .
  • . Since , there are -many linear systems on X, the one given by monodromy representation
    by sending
  • Horizontal sections of vector bundles with a flat connection. If is a vector bundle with flat connection , then

    is a local system.

    For instance, take and the trivial bundle. Sections of E are n-tuples of functions on X, so defines a flat connection on E, as does for any matrix of one-forms on X. The horizontal sections are then

    i.e., the solutions to the linear differential equation .

    If extends to a one-form on the above will also define a local system on , so will be trivial since . So to give an interesting example, choose one with a pole at 0:

    in which case for ,

  • An n-sheeted covering map is a local system with sections locally the set . Similarly, a fibre bundle with discrete fibre is a local system, because each path lifts uniquely to a given lift of its basepoint. (The definition adjusts to include set-valued local systems in the obvious way).
  • A local system of k-vector spaces on X is the same as a k-linear representation of the group .
  • If X is a variety, local systems are the same thing as D-modules that are in addition coherent as O-modules.

If the connection is not flat, parallel transporting a fibre around a contractible loop at x may give a nontrivial automorphism of the fibre at the base point x, so there is no chance to define a locally constant sheaf this way.

The Gauss–Manin connection is a very interesting example of a connection, whose horizontal sections occur in the study of variation of Hodge structures.

Generalization[edit]

Local systems have a mild generalization to constructible sheaves. A constructible sheaf on a locally path connected topological space is a sheaf such that there exists a stratification of

where is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map . For example, if we look at the complex points of the morphism

then the fibers over

are the smooth plane curve given by , but the fibers over are . If we take the derived pushforward then we get a constructible sheaf. Over we have the local systems

while over we have the local systems

where is the genus of the plane curve (which is ).

Applications[edit]

The cohomology with local coefficients in the module corresponding to the orientation covering can be used to formulate Poincaré duality for non-orientable manifolds: see Twisted Poincaré duality.

See also[edit]

References[edit]

  1. ^ Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics. 44 (4): 610–627. doi:10.2307/1969099. MR 0009114.

External links[edit]