Local system

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

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 ${\mathcal {L}}$ is a local system if every point has an open neighborhood $U$ such that ${\mathcal {L}}|_{U}$ is a constant sheaf.

Equivalent definitions

Path-connected spaces

If X is path-connected, a local system ${\mathcal {L}}$ 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

\rho :\pi _{1}(X,x)\to {\text{Aut}}(L)

and similarly for local systems of modules,... The map $\pi _{1}(X,x)\to {\text{End}}(L)$ giving the local system ${\mathcal {L}}$ is called the monodromy representation of ${\mathcal {L}}$ .

Proof of equivalence

Take local system ${\mathcal {L}}$ and a loop $\gamma$ at x. It's easy to show that any local system on $[0,1]$ is constant. For instance, $\gamma ^{*}{\mathcal {L}}$ is constant. This gives an isomorphism $(\gamma ^{*}{\mathcal {L}})_{0}\simeq \Gamma ([0,1],{\mathcal {L}})\simeq (\gamma ^{*}{\mathcal {L}})_{1}$ , i.e. between L and itself. Conversely, given a homomorphism $\rho :\pi _{1}(X,x)\to {\text{End}}(L)$ , consider the constant sheaf ${\underline {L}}$ on the universal cover ${\widetilde {X}}$ of X. The deck-transform-invariant sections of ${\underline {L}}$ 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

{\mathcal {L}}(\rho )_{U}\ =\ \left\{{\text{sections }}s\in {\underline {L}}_{\pi ^{-1}(U)}{\text{ with }}\theta \circ s=\rho (\theta )s{\text{ for all }}\theta \in {\text{ Deck}}({\widetilde {X}}/X)=\pi _{1}(X,x)\right\}

where $\pi :{\widetilde {X}}\to X$ 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

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

{\mathcal {L}}\colon \Pi _{1}(X)\to {\textbf {Mod}}(R)

from the fundamental groupoid of $X$ to the category of modules over a commutative ring $R$ . Typically $R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C}$ . What this is saying is that at every point $x\in X$ we should assign a module $M$ with a representations of $\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)$ such that these representations are compatible with change of basepoint $x\to y$ for the fundamental group.

Examples

Constant sheaves. For instance, ${\underline {\mathbb {Q} }}_{X}$ . This is a useful tool for computing cohomology since the sheaf cohomology

H^{k}(X,{\underline {\mathbb {Q} }}_{X})\cong H_{\text{sing}}^{k}(X,\mathbb {Q} )

is isomorphic to the singular cohomology of

X

.

$X=\mathbb {R} ^{2}\setminus \{(0,0)\}$ . Since $\pi _{1}(\mathbb {R} ^{2}\setminus \{(0,0)\})=\mathbb {Z}$ , there are $S^{1}$ -many linear systems on X, the $\theta \in \mathbb {R}$ one given by monodromy representation

\pi _{x}(X)=\mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )

by sending

n\mapsto e^{in\theta }

Horizontal sections of vector bundles with a flat connection. If $E\to X$ is a vector bundle with flat connection $\nabla$ , then

E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}

is a local system.

For instance, take

X=\mathbb {C} \setminus 0

and

E=X\times \mathbb {C} ^{n}

the trivial bundle. Sections of E are n-tuples of functions on X, so

\nabla _{0}(f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})

defines a flat connection on E, as does

\nabla (f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})-\Theta (x)(f_{1},\dots ,f_{n})^{t}

for any matrix of one-forms

\Theta

on X. The horizontal sections are then

E_{U}^{\nabla }=\left\{(f_{1},\dots ,f_{n})\in E_{U}:(df_{1},\dots ,df_{n})=\Theta (f_{1},\dots ,f_{n})^{t}\right\}

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

df_{i}=\sum \Theta _{ij}f_{j}

.

If

\Theta

extends to a one-form on

\mathbb {C}

the above will also define a local system on

\mathbb {C}

, so will be trivial since

\pi _{1}(\mathbb {C} )=0

. So to give an interesting example, choose one with a pole at 0:

\Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}

in which case for

\nabla =d+\Theta

,

E_{U}^{\nabla }=\left\{f_{1},f_{2}:U\to \mathbb {C} \ \ {\text{ with }}f'_{1}=f_{2}/x\ \ f_{2}'=f_{1}+e^{x}f_{2}\right\}

An n-sheeted covering map $X\to Y$ is a local system with sections locally the set $\{1,\dots ,n\}$ . 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 $\pi _{1}(X,x)$ .
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

Local systems have a mild generalization to constructible sheaves. A constructible sheaf on a locally path connected topological space $X$ is a sheaf ${\mathcal {L}}$ such that there exists a stratification of

X=\coprod X_{\lambda }

where ${\mathcal {L}}|_{X_{\lambda }}$ is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map $f:X\to Y$ . For example, if we look at the complex points of the morphism

f:X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x,y,z]}{(stf(x,y,z))}}\right)\to {\text{Spec}}(\mathbb {C} [s,t])

then the fibers over

\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)

are the smooth plane curve given by $f$ , but the fibers over $\mathbb {V}$ are $\mathbb {P} ^{2}$ . If we take the derived pushforward $\mathbf {R} f_{!}({\underline {\mathbb {Q} }}_{X})$ then we get a constructible sheaf. Over $\mathbb {V}$ we have the local systems

{\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{4}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {V} (st)}&={\underline {0}}_{\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}}

while over $\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)$ we have the local systems

{\begin{aligned}\mathbf {R} ^{0}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{1}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}^{\oplus 2g}\\\mathbf {R} ^{2}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {\mathbb {Q} }}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}\\\mathbf {R} ^{k}f_{!}({\underline {\mathbb {Q} }}_{X})|_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}&={\underline {0}}_{\mathbb {A} _{s,t}^{2}-\mathbb {V} (st)}{\text{ otherwise}}\end{aligned}}

where $g$ is the genus of the plane curve (which is $g=({\text{deg}}(f)-1)({\text{deg}}(f)-2)/2$ ).

Applications

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.

Topological Bivariant Theory

For maps $f\colon X\to Y$ there is a Bivariant theory similar to William Fulton's, called Topological Bivariant Theory.[2] Defining such a theory requires local systems and the six-functor formalism. Bivariant theories are characterized by the property

$H^{i}(X{\xrightarrow {f}}Y)={\text{Hom}}_{D^{b}(Y)}(\mathbb {R} f_{!}\mathbb {Q} _{X},\mathbb {Q} _{Y}[+i])$

For example, this can be computed in some simple cases. If $Y$ is a point, this recovers Borel–Moore homology. If $Y=X$ and the map is the identity, then this is oridinary cohomology. Another informative class of example includes covering spaces. For example, if $f\colon \mathbb {G} _{m}\to \mathbb {G} _{m}=\mathbb {C} ^{*}$ is the degree $d$ covering given by $z\mapsto z^{d}$ . Then, at the stalk level the cohomology groups are of the form

$\mathbb {R} f_{!}(\mathbb {Q} _{\mathbb {G} _{m}})|_{x}\cong \mathbb {Q} ^{\oplus d}$

and the monodromy for $\pi _{1}(\mathbb {G} _{m})=\mathbb {Z}$ is given by the map taking a branch to its next branch and the $d$ -th branch to the first branch. That is, $\phi \colon \mathbb {Z} \to {\text{Aut}}(\mathbb {Q} ^{d})$ is generated by the matrix

$\phi (1)={\begin{pmatrix}0&0&0&\cdots &0&1\\1&0&0&\cdots &0&0\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\0&0&0&\cdots &1&0\end{pmatrix}}$

References

Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics. 44 (4): 610–627. doi:10.2307/1969099. MR 0009114.
de Cataldo, Mark Andrea A.; Migliorini, Luca. "The Gysin map is compatible with mixed Hodge Structures" (PDF). Archived (PDF) from the original on 16 February 2020.

External links

"What local system really is". Stack Exchange.
Schnell, Christian. "Computing Cohomology of Local Systems" (PDF). Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.
Williamson, Geordie. "An illustrated guide to perverse sheaves" (PDF).
MacPherson, Robert (December 15, 1990). "Intersection homology and perverse sheaves" (PDF).
El Zein, Fouad; Snoussi, Jawad. "Local systems and constructible sheaves" (PDF).

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

[2] Cataldo, Mark Andrea A.; Migliorini, Luca. "The Gysin map is compatible with mixed Hodge Structures" (PDF). Archived (PDF) from the original on 16 February 2020.