Local system

In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. ^[1]

Local systems are the building blocks of more general tools, such as constructible and perverse sheaves.

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 ${\displaystyle {\mathcal {L}}}$ is a local system if every point has an open neighborhood ${\displaystyle U}$ such that the restricted sheaf ${\displaystyle {\mathcal {L}}|_{U}}$ is isomorphic to the sheafification of some constant presheaf. ^{[ clarification needed ]}

Equivalent definitions

Path-connected spaces

If X is path-connected,^{[ clarification needed ]} a local system ${\displaystyle {\mathcal {L}}}$ of abelian groups has the same stalk L at every point. There is a bijective correspondence between local systems on X and group homomorphisms

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

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

Proof of equivalence

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

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

where ${\displaystyle \pi$ :{\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.

This correspondence can be upgraded to an equivalence of categories between the category of local systems of abelian groups on X and the category of abelian groups endowed with an action of ${\displaystyle \pi _{1}(X,x)}$ (equivalently, ${\displaystyle \mathbb {Z} [\pi _{1}(X,x)]}$-modules). ^[2]

Stronger definition on non-connected spaces

A stronger nonequivalent definition that works for non-connected X is: the following: a local system is a covariant functor

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

from the fundamental groupoid of ${\displaystyle X}$ to the category of modules over a commutative ring ${\displaystyle R}$, where typically ${\displaystyle R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }$. This is equivalently the data of an assignment to every point ${\displaystyle x\in X}$ a module ${\displaystyle M}$ along with a group representation ${\displaystyle \rho _{x}:\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)}$ such that the various ${\displaystyle \rho _{x}}$ are compatible with change of basepoint ${\displaystyle x\to y}$ and the induced map ${\displaystyle \pi _{1}(X,x)\to \pi _{1}(X,y)}$ on fundamental groups.

Examples

• Constant sheaves such as ${\displaystyle {\underline {\mathbb {Q} }}_{X}}$. This is a useful tool for computing cohomology since in good situations, there is an isomorphism between sheaf cohomology and singular cohomology:

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

• Let ${\displaystyle X=\mathbb {R} ^{2}\setminus \{(0,0)\}}$. Since ${\displaystyle \pi _{1}(\mathbb {R} ^{2}\setminus \{(0,0)\})=\mathbb {Z} }$, there is an ${\displaystyle S^{1}}$ family of local systems on X corresponding to the maps ${\displaystyle n\mapsto e^{in\theta }}$:

$${\displaystyle \rho _{\theta }:\pi _{1}(X;x_{0})\cong \mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )}$$

• Horizontal sections of vector bundles with a flat connection. If ${\displaystyle E\to X}$ is a vector bundle with flat connection ${\displaystyle \nabla }$, then there is a local system given by
  $${\displaystyle E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}}$$
  For instance, take ${\displaystyle X=\mathbb {C} \setminus 0}$ and ${\displaystyle E=X\times \mathbb {C} ^{n}}$, the trivial bundle. Sections of E are n-tuples of functions on X, so ${\displaystyle \nabla _{0}(f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})}$ defines a flat connection on E, as does ${\displaystyle \nabla (f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})-\Theta (x)(f_{1},\dots ,f_{n})^{t}}$ for any matrix of one-forms ${\displaystyle \Theta }$ on X. The horizontal sections are then

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

i.e., the solutions to the linear differential equation ${\displaystyle df_{i}=\sum \Theta _{ij}f_{j}}$.

If ${\displaystyle \Theta }$ extends to a one-form on ${\displaystyle \mathbb {C} }$ the above will also define a local system on ${\displaystyle \mathbb {C} }$, so will be trivial since ${\displaystyle \pi _{1}(\mathbb {C} )=0}$. So to give an interesting example, choose one with a pole at 0:

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

in which case for ${\displaystyle \nabla =d+\Theta }$,

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

• An n-sheeted covering map ${\displaystyle X\to Y}$ is a local system with fibers given by the set ${\displaystyle \{1,\dots ,n\}}$. 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 equivalent to a k-linear representation of ${\displaystyle \pi _{1}(X,x)}$.

• If X is a variety, local systems are the same thing as D-modules which are additionally coherent O_X-modules (see O modules).

• If the connection is not flat (i.e. its curvature is nonzero), then parallel transport of a fibre F_x over x around a contractible loop based at x_0 may give a nontrivial automorphism of F_x, so locally constant sheaves can not necessarily be defined for non-flat connections.

• The Gauss–Manin connection is a prominent example of a connection whose horizontal sections are studied in relation to variation of Hodge structures.

Cohomology

There are several ways to define the cohomology of a local system, called cohomology with local coefficients, which become equivalent under mild assumptions on X.

• Given a locally constant sheaf ${\displaystyle {\mathcal {L}}}$ of abelian groups on X, we have the sheaf cohomology groups ${\displaystyle H^{j}(X,{\mathcal {L}})}$ with coefficients in ${\displaystyle {\mathcal {L}}}$.

• Given a locally constant sheaf ${\displaystyle {\mathcal {L}}}$ of abelian groups on X, let ${\displaystyle C^{n}(X;{\mathcal {L}})}$ be the group of all functions f which map each singular n-simplex ${\displaystyle \sigma \colon \Delta ^{n}\to X}$ to a global section ${\displaystyle f(\sigma )}$ of the inverse-image sheaf ${\displaystyle \sigma ^{-1}{\mathcal {L}}}$. These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define ${\displaystyle H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})}$ to be the cohomology of this complex.

• The group ${\displaystyle C_{n}({\widetilde {X}})}$ of singular n-chains on the universal cover of X has an action of ${\displaystyle \pi _{1}(X,x)}$ by deck transformations. Explicitly, a deck transformation ${\displaystyle \gamma \colon {\widetilde {X}}\to {\widetilde {X}}}$ takes a singular n-simplex ${\displaystyle \sigma \colon \Delta ^{n}\to {\widetilde {X}}}$ to ${\displaystyle \gamma \circ \sigma }$. Then, given an abelian group L equipped with an action of ${\displaystyle \pi _{1}(X,x)}$, one can form a cochain complex from the groups ${\displaystyle \operatorname {Hom} _{\pi _{1}(X,x)}(C_{n}({\widetilde {X}}),L)}$ of ${\displaystyle \pi _{1}(X,x)}$-equivariant homomorphisms as above. Define ${\displaystyle H_{\mathrm {sing} }^{j}(X;L)}$ to be the cohomology of this complex.

If X is paracompact and locally contractible, then ${\displaystyle H^{j}(X,{\mathcal {L}})\cong H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})}$. ^[3] If ${\displaystyle {\mathcal {L}}}$ is the local system corresponding to L, then there is an identification ${\displaystyle C^{n}(X;{\mathcal {L}})\cong \operatorname {Hom} _{\pi _{1}(X,x)}(C_{n}({\widetilde {X}}),L)}$ compatible with the differentials, ^[4] so ${\displaystyle H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})\cong H_{\mathrm {sing} }^{j}(X;L)}$.

Generalization

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

${\displaystyle X=\coprod X_{\lambda }}$

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

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

then the fibers over

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

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

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

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

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

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

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.

See also

• Leray spectral sequence
• Gauss–Manin connection
• D-module
• Intersection homology
• Perverse sheaf

References

1. Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics . 44 (4): 610–627. doi:10.2307/1969099. MR   0009114.
2. Milne, James S. (2017). Introduction to Shimura Varieties . Proposition 14.7.
3. Bredon, Glen E. (1997). Sheaf Theory, Second Edition, Graduate Texts in Mathematics, vol. 25, Springer-Verlag. Chapter III, Theorem 1.1.
4. Hatcher, Allen (2001). Algebraic Topology, Cambridge University Press. Section 3.H.

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).