Local system

Let X be a topological space. A local system (of abelian groups/modules...) on X is a locally constant sheaf (of abelian groups/of modules...) on X. In other words, a sheaf $\backslash mathcal\{L\}$ is a local system if every point has an open neighborhood $U$ such that the restricted sheaf $\backslash mathcal\{L\}|\_U$ is isomorphic to the sheafification of some constant presheaf. {{clarify| there is some ambiguity of constant sheaf, some uses the definition that restriction is identity; others uses the convention that isomorphic to such a sheaf suffices.|date=September 2022}}

If X is path-connected,{{clarify|Don't you also need "locally simply connected?"|date=August 2022}} a local system $\backslash mathcal\{L\}$ of abelian groups has the same stalk $L$ at every point. There is a bijective correspondence between local systems on X and group homomorphisms

: $\backslash rho:\; \backslash pi\_1(X,x)\; \backslash to\; \backslash text\{Aut\}(L)$

and similarly for local systems of modules. The map $\backslash pi\_1(X,x)\; \backslash to\; \backslash text\{Aut\}(L)$ giving the local system $\backslash mathcal\{L\}$ is called the monodromy representation of $\backslash mathcal\{L\}$.

{{math proof|title=Proof of equivalence|proof=Take local system $\backslash mathcal\{L\}$ and a loop $\backslash gamma$ at x. It's easy to show that any local system on $[0,1]$ is constant. For instance, $\backslash gamma^*\; \backslash mathcal\{L\}$ is constant. This gives an isomorphism $(\backslash gamma^*\backslash mathcal\{L\})\_0\backslash simeq\; \backslash Gamma([0,1],\; \backslash mathcal\{L\})\; \backslash simeq\; (\backslash gamma^*\backslash mathcal\{L\})\_1$, i.e. between $L$ and itself.

Conversely, given a homomorphism $\backslash rho:\; \backslash pi\_1(X,x)\backslash to\; \backslash text\{Aut\}(L)$, consider the constant sheaf $\backslash underline\{L\}$ on the universal cover $\backslash widetilde\{X\}$ of X. The deck-transform-invariant sections of $\backslash underline\{L\}$ 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

: $\backslash mathcal\{L\}(\backslash rho)\_U\backslash \; =\; \backslash \; \backslash left\backslash \{\; \backslash text\{sections\; \}s\; \backslash in\; \backslash underline\{L\}\_\{\backslash pi^\{-1\}(U)\}\; \backslash text\{\; with\; \}\backslash theta\backslash circ\; s=\backslash rho(\backslash theta)\; s\; \backslash text\{\; for\; all\; \}\backslash theta\; \backslash in\backslash text\{\; Deck\}(\backslash widetilde\{X\}/X)=\backslash pi\_1(X,x)\; \backslash right\backslash \}$

where $\backslash pi:\backslash widetilde\{X\}\backslash to\; X$ 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 $\backslash pi\_1(X,x)$ (equivalently, $\backslash mathbb\{Z\}[\backslash pi\_1(X,x)]$-modules). Milne, James S. (2017). [https://www.jmilne.org/math/xnotes/svi.pdf Introduction to Shimura Varieties]. Proposition 14.7.

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

:$\backslash mathcal\{L\}\backslash colon\; \backslash Pi\_1(X)\; \backslash to\; \backslash textbf\{Mod\}(R)$

from the fundamental groupoid of $X$ to the category of modules over a commutative ring $R$, where typically $R\; =\; \backslash Q,\backslash R,\backslash Complex$. This is equivalently the data of an assignment to every point $x\backslash in\; X$ a module $M$ along with a group representation $\backslash rho\_x:\; \backslash pi\_1(X,x)\; \backslash to\; \backslash text\{Aut\}\_R(M)$ such that the various $\backslash rho\_x$ are compatible with change of basepoint $x\; \backslash to\; y$ and the induced map $\backslash pi\_1(X,\; x)\; \backslash to\; \backslash pi\_1(X,\; y)$ on fundamental groups.

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

$$H^k(X,\backslash underline\{\backslash Q\}\_X)\; \backslash cong\; H^k\_\backslash text\{sing\}(X,\backslash Q)$$

• Let $X=\backslash R^2\; \backslash setminus\; \backslash \{(0,0)\backslash \}$. Since $\backslash pi\_1(\backslash R^2\; \backslash setminus\; \backslash \{(0,0)\backslash \})=\backslash mathbb\{Z\}$, there is an $S^1$ family of local systems on X corresponding to the maps $n\; \backslash mapsto\; e^\{in\backslash theta\}$:

$$\backslash rho\_\backslash theta:\; \backslash pi\_1(X;\; x\_0)\; \backslash cong\; \backslash Z\; \backslash to\; \backslash text\{Aut\}\_\backslash Complex(\backslash Complex)$$

• Horizontal sections of vector bundles with a flat connection. If $E\backslash to\; X$ is a vector bundle with flat connection $\backslash nabla$, then there is a local system given by $$E^\backslash nabla\_U=\backslash left\backslash \{\backslash text\{sections\; \}s\backslash in\; \backslash Gamma(U,E)\; \backslash text\{\; which\; are\; horizontal:\; \}\backslash nabla\; s=0\backslash right\backslash \}$$ For instance, take $X=\backslash Complex\; \backslash setminus\; 0$ and $E\; =\; X\; \backslash times\; \backslash Complex^n$, the trivial bundle. Sections of E are n-tuples of functions on X, so $\backslash nabla\_0(f\_1,\backslash dots,f\_n)=\; (df\_1,\backslash dots,df\_n)$ defines a flat connection on E, as does $\backslash nabla(f\_1,\backslash dots,f\_n)=\; (df\_1,\backslash dots,df\_n)-\backslash Theta(x)(f\_1,\backslash dots,f\_n)^t$ for any matrix of one-forms $\backslash Theta$ on X. The horizontal sections are then
  $$E^\backslash nabla\_U=\; \backslash left\backslash \{(f\_1,\backslash dots,f\_n)\backslash in\; E\_U:\; (df\_1,\backslash dots,df\_n)=\backslash Theta\; (f\_1,\backslash dots,f\_n)^t\backslash right\backslash \}$$ i.e., the solutions to the linear differential equation $df\_i\; =\; \backslash sum\; \backslash Theta\_\{ij\}\; f\_j$.

  If $\backslash Theta$ extends to a one-form on $\backslash Complex$ the above will also define a local system on $\backslash Complex$, so will be trivial since $\backslash pi\_1(\backslash Complex)\; =\; 0$. So to give an interesting example, choose one with a pole at 0:

  in which case for $\backslash nabla=\; d+\; \backslash Theta$, $$E^\backslash nabla\_U\; =\backslash left\backslash \{\; f\_1,f\_2:\; U\; \backslash to\; \backslash mathbb\{C\}\; \backslash \; \backslash \; \backslash text\{\; with\; \}\; f\text{'}\_1=\; f\_2/x\; \backslash \; \backslash \; f\_2\text{'}=f\_1+\; e^x\; f\_2\backslash right\backslash \}$$

• An n-sheeted covering map $X\backslash to\; Y$ is a local system with fibers given by the set $\backslash \{1,\backslash dots,n\backslash \}$. 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 $\backslash pi\_1(X,x)$.

• 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.

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 $\backslash mathcal\{L\}$ of abelian groups on X, we have the sheaf cohomology groups $H^j(X,\backslash mathcal\{L\})$ with coefficients in $\backslash mathcal\{L\}$.

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

• The group $C\_n(\backslash widetilde\{X\})$ of singular n-chains on the universal cover of X has an action of $\backslash pi\_1(X,x)$ by deck transformations. Explicitly, a deck transformation $\backslash gamma\backslash colon\backslash widetilde\{X\}\backslash to\backslash widetilde\{X\}$ takes a singular n-simplex $\backslash sigma\backslash colon\backslash Delta^n\backslash to\backslash widetilde\{X\}$ to $\backslash gamma\backslash circ\backslash sigma$. Then, given an abelian group L equipped with an action of $\backslash pi\_1(X,x)$, one can form a cochain complex from the groups $\backslash operatorname\{Hom\}\_\{\backslash pi\_1(X,x)\}(C\_n(\backslash widetilde\{X\}),L)$ of $\backslash pi\_1(X,x)$-equivariant homomorphisms as above. Define $H^j\_\backslash mathrm\{sing\}(X;L)$ to be the cohomology of this complex.

If X is paracompact and locally contractible, then $H^j(X,\backslash mathcal\{L\})\backslash cong\; H^j\_\backslash mathrm\{sing\}(X;\backslash mathcal\{L\})$. Bredon, Glen E. (1997). Sheaf Theory, Second Edition, Graduate Texts in Mathematics, vol. 25, Springer-Verlag. Chapter III, Theorem 1.1. If $\backslash mathcal\{L\}$ is the local system corresponding to L, then there is an identification $C^n(X;\backslash mathcal\{L\})\backslash cong\backslash operatorname\{Hom\}\_\{\backslash pi\_1(X,x)\}(C\_n(\backslash widetilde\{X\}),L)$ compatible with the differentials, Hatcher, Allen (2001). Algebraic Topology, Cambridge University Press. Section 3.H. so $H^j\_\backslash mathrm\{sing\}(X;\backslash mathcal\{L\})\backslash cong\; H^j\_\backslash mathrm\{sing\}(X;L)$.

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

:$X\; =\; \backslash coprod\; X\_\backslash lambda$

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

:$f:X\; =\; \backslash text\{Proj\}\backslash left(\backslash frac\{\backslash Complex[s,t][x,y,z]\}\{(st\backslash cdot\; h(x,y,z))\}\backslash right)\; \backslash to\; \backslash text\{Spec\}(\backslash Complex[s,t])$

then the fibers over

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

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

:$$

\begin{align}

\mathbf{R}^0f_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{\Q}_{\mathbb{V}(st)} \\

\mathbf{R}^2f_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{\Q}_{\mathbb{V}(st)} \\

\mathbf{R}^4f_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{\Q}_{\mathbb{V}(st)} \\

\mathbf{R}^kf_!(\underline{\mathbb{Q}}_X)|_{\mathbb{V}(st)} &= \underline{0}_{\mathbb{V}(st)} \text{ otherwise}

\end{align}

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

:$\backslash begin\{align\}$

\mathbf{R}^0f_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{\Q}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} \\

\mathbf{R}^1f_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{\Q}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)}^{\oplus 2g} \\

\mathbf{R}^2f_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{\Q}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} \\

\mathbf{R}^kf_!(\underline{\Q}_X)|_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} &= \underline{0}_{\mathbb{A}^2_{s,t} - \mathbb{V}(st)} \text{ otherwise}

\end{align}

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

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.

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

{{reflist}}

• {{cite web|url=https://math.stackexchange.com/q/13332|title=What local system really is|publisher= Stack Exchange}}
• {{cite web|url=https://www.math.stonybrook.edu/~cschnell/pdf/notes/locsys.pdf|title= Computing Cohomology of Local Systems|first=Christian|last=Schnell}} Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.
• {{cite web|url=http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf|title= An illustrated guide to perverse sheaves|first= Geordie|last= Williamson|author-link= Geordie Williamson}}
• {{cite web|url=http://faculty.tcu.edu/gfriedman/notes/ih.pdf |title=Intersection homology and perverse sheaves| first=Robert | last=MacPherson | author-link=Robert MacPherson (mathematician)|date=December 15, 1990}}
• {{cite web|url=https://webusers.imj-prg.fr/~fouad.elzein/elzein-snoussif.pdf|title= Local systems and constructible sheaves|first1= Fouad|last1= El Zein|first2= Jawad |last2=Snoussi}}

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Category:Sheaf theory

Category:Algebraic topology