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<h2 id="definition">Definition</h2> <p>Let <i>X</i> be a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>. A local system (of <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a>/<a href="/facts/Module_over_a_ring/jP0LIhFL">modules</a>...) on <i>X</i> is a <a href="/facts/Locally_constant_sheaf/mW64LDW9">locally constant sheaf</a> (of abelian groups/<a href="/facts/Sheaf_of_modules/grSMBXho">of modules</a>...) on <i>X</i>. In other words, a sheaf L {\displaystyle {\mathcal {L}}} is a local system if every point has an open neighborhood U {\displaystyle U} such that the restricted sheaf L | U {\displaystyle {\mathcal {L}}|_{U}} is isomorphic to the sheafification of some constant presheaf. </p> <h3>Equivalent definitions</h3> <h4>Path-connected spaces</h4> <p>If <i>X</i> is <a href="/facts/Connected_space/5CUTxfoA">path-connected</a>, a local system L {\displaystyle {\mathcal {L}}} of abelian groups has the same stalk L {\displaystyle L} at every point. There is a bijective correspondence between local systems on <i>X</i> and group homomorphisms </p> ρ : π 1 ( X , x ) → Aut ( L ) {\displaystyle \rho :\pi _{1}(X,x)\to {\text{Aut}}(L)} <p>and similarly for local systems of modules. The map π 1 ( X , x ) → Aut ( L ) {\displaystyle \pi _{1}(X,x)\to {\text{Aut}}(L)} giving the local system L {\displaystyle {\mathcal {L}}} is called the <a href="/facts/Monodromy/ldkpxExM">monodromy</a> representation of L {\displaystyle {\mathcal {L}}} . </p> <i>Proof of equivalence</i> <p>Take local system L {\displaystyle {\mathcal {L}}} and a loop γ {\displaystyle \gamma } at <i>x</i>. It's easy to show that any local system on [ 0 , 1 ] {\displaystyle [0,1]} is constant. For instance, γ ∗ L {\displaystyle \gamma ^{*}{\mathcal {L}}} is constant. This gives an isomorphism ( γ ∗ L ) 0 ≃ Γ ( [ 0 , 1 ] , L ) ≃ ( γ ∗ L ) 1 {\displaystyle (\gamma ^{*}{\mathcal {L}})_{0}\simeq \Gamma ([0,1],{\mathcal {L}})\simeq (\gamma ^{*}{\mathcal {L}})_{1}} , i.e. between L {\displaystyle L} and itself. Conversely, given a homomorphism ρ : π 1 ( X , x ) → Aut ( L ) {\displaystyle \rho :\pi _{1}(X,x)\to {\text{Aut}}(L)} , consider the <i>constant</i> sheaf L _ {\displaystyle {\underline {L}}} on the universal cover X ~ {\displaystyle {\widetilde {X}}} of <i>X</i>. The deck-transform-invariant sections of L _ {\displaystyle {\underline {L}}} gives a local system on <i>X</i>. Similarly, the deck-transform-<i>ρ</i>-equivariant sections give another local system on <i>X</i>: for a small enough open set <i>U</i>, it is defined as </p> L ( ρ ) U = { sections s ∈ L _ π − 1 ( U ) with θ ∘ s = ρ ( θ ) s for all θ ∈ Deck ( X ~ / X ) = π 1 ( X , x ) } {\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\}} <p>where π : X ~ → X {\displaystyle \pi :{\widetilde {X}}\to X} is the universal covering. </p> <p>This shows that (for <i>X</i> path-connected) a local system is precisely a sheaf whose pullback to the <a href="/facts/Universal_cover/bDRGasl7">universal cover</a> of <i>X</i> is a constant sheaf. </p><p>This correspondence can be upgraded to an equivalence of categories between the category of local systems of abelian groups on <i>X</i> and the category of abelian groups endowed with an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} (equivalently, Z [ π 1 ( X , x ) ] {\displaystyle \mathbb {Z} [\pi _{1}(X,x)]} -modules).<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h4>Stronger definition on non-connected spaces</h4> <p>A stronger nonequivalent definition that works for non-connected <i>X</i> is the following: a local system is a <a href="/facts/Covariant_functor/l2u3rIBE">covariant functor</a> </p> L : Π 1 ( X ) → Mod ( R ) {\displaystyle {\mathcal {L}}\colon \Pi _{1}(X)\to {\textbf {Mod}}(R)} <p>from the fundamental groupoid of X {\displaystyle X} to the category of modules over a commutative ring R {\displaystyle R} , where typically R = Q , R , C {\displaystyle R=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} } . This is equivalently the data of an assignment to every point x ∈ X {\displaystyle x\in X} a module M {\displaystyle M} along with a group representation ρ x : π 1 ( X , x ) → Aut R ( M ) {\displaystyle \rho _{x}:\pi _{1}(X,x)\to {\text{Aut}}_{R}(M)} such that the various ρ x {\displaystyle \rho _{x}} are compatible with change of basepoint x → y {\displaystyle x\to y} and the induced map π 1 ( X , x ) → π 1 ( X , y ) {\displaystyle \pi _{1}(X,x)\to \pi _{1}(X,y)} on <a href="/facts/Fundamental_groups/VyYtLqSP">fundamental groups</a>. </p> <h2 id="examples">Examples</h2> <ul><li><a href="/facts/Constant_sheaf/RfrbWp81"> Constant sheaves</a> such as Q _ X {\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:</li></ul> <p> H k ( X , Q _ X ) ≅ H sing k ( X , Q ) {\displaystyle H^{k}(X,{\underline {\mathbb {Q} }}_{X})\cong H_{\text{sing}}^{k}(X,\mathbb {Q} )} </p> <ul><li>Let X = R 2 ∖ { ( 0 , 0 ) } {\displaystyle X=\mathbb {R} ^{2}\setminus \{(0,0)\}} . Since π 1 ( R 2 ∖ { ( 0 , 0 ) } ) = Z {\displaystyle \pi _{1}(\mathbb {R} ^{2}\setminus \{(0,0)\})=\mathbb {Z} } , there is an S 1 {\displaystyle S^{1}} family of local systems on <i>X</i> corresponding to the maps n ↦ e i n θ {\displaystyle n\mapsto e^{in\theta }} :</li></ul> <p> ρ θ : π 1 ( X ; x 0 ) ≅ Z → Aut C ( C ) {\displaystyle \rho _{\theta }:\pi _{1}(X;x_{0})\cong \mathbb {Z} \to {\text{Aut}}_{\mathbb {C} }(\mathbb {C} )} </p> <ul><li>Horizontal sections of vector bundles with a flat connection. If E → X {\displaystyle E\to X} is a vector bundle with flat connection ∇ {\displaystyle \nabla } , then there is a local system given by E U ∇ = { sections s ∈ Γ ( U , E ) which are horizontal: ∇ s = 0 } {\displaystyle E_{U}^{\nabla }=\left\{{\text{sections }}s\in \Gamma (U,E){\text{ which are horizontal: }}\nabla s=0\right\}} For instance, take X = C ∖ 0 {\displaystyle X=\mathbb {C} \setminus 0} and E = X × C n {\displaystyle E=X\times \mathbb {C} ^{n}} , the trivial bundle. Sections of <i>E</i> are <i>n</i>-tuples of functions on <i>X</i>, so ∇ 0 ( f 1 , … , f n ) = ( d f 1 , … , d f n ) {\displaystyle \nabla _{0}(f_{1},\dots ,f_{n})=(df_{1},\dots ,df_{n})} defines a flat connection on <i>E</i>, as does ∇ ( f 1 , … , f n ) = ( d f 1 , … , d f n ) − Θ ( x ) ( f 1 , … , f n ) t {\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 Θ {\displaystyle \Theta } on <i>X</i>. The horizontal sections are then E U ∇ = { ( f 1 , … , f n ) ∈ E U : ( d f 1 , … , d f n ) = Θ ( f 1 , … , f n ) t } {\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 d f i = ∑ Θ i j f j {\displaystyle df_{i}=\sum \Theta _{ij}f_{j}} .<p>If Θ {\displaystyle \Theta } extends to a one-form on C {\displaystyle \mathbb {C} } the above will also define a local system on C {\displaystyle \mathbb {C} } , so will be trivial since π 1 ( C ) = 0 {\displaystyle \pi _{1}(\mathbb {C} )=0} . So to give an interesting example, choose one with a pole at <i>0</i>:</p> Θ = ( 0 d x / x d x e x d x ) {\displaystyle \Theta ={\begin{pmatrix}0&dx/x\\dx&e^{x}dx\end{pmatrix}}} in which case for ∇ = d + Θ {\displaystyle \nabla =d+\Theta } , E U ∇ = { f 1 , f 2 : U → C with f 1 ′ = f 2 / x f 2 ′ = f 1 + e x f 2 } {\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\}} </li></ul> <ul><li>An <i>n</i>-sheeted covering map X → Y {\displaystyle X\to Y} is a local system with fibers given by the set { 1 , … , n } {\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).</li></ul> <ul><li>A local system of <i>k</i>-vector spaces on <i>X</i> is equivalent to a <i>k</i>-linear <a href="/facts/Group_representation/FsuuDagQ">representation</a> of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} .</li></ul> <ul><li>If <i>X</i> is a variety, local systems are the same thing as <a href="/facts/D-modules/l1y6evYe">D-modules</a> which are additionally coherent <i>O_X</i>-modules (see <a href="/facts/Sheaf_of_modules/grSMBXho"> O modules</a>).</li></ul> <ul><li>If the connection is not flat (i.e. its <a href="/facts/Curvature/RFCAEsj8">curvature</a> is nonzero), then parallel transport of a fibre <i>F_x</i> over <i>x</i> around a contractible loop based at <i>x</i>_0 may give a nontrivial automorphism of <i>F_x</i>, so locally constant sheaves can not necessarily be defined for non-flat connections.</li></ul> <ul><li>The <a href="/facts/Gauss%E2%80%93Manin_connection/wy50u69z">Gauss–Manin connection</a> is a prominent example of a connection whose horizontal sections are studied in relation to <a href="/facts/Hodge_structure/mudXG4sQ">variation of Hodge structures</a>.</li></ul> <h2 id="cohomology">Cohomology</h2> <p>There are several ways to define the cohomology of a local system, called cohomology with local coefficients, which become equivalent under mild assumptions on <i>X</i>. </p> <ul><li>Given a locally constant sheaf L {\displaystyle {\mathcal {L}}} of abelian groups on <i>X</i>, we have the <a href="/facts/Sheaf_cohomology/TrcpmD6k">sheaf cohomology</a> groups H j ( X , L ) {\displaystyle H^{j}(X,{\mathcal {L}})} with coefficients in L {\displaystyle {\mathcal {L}}} .</li></ul> <ul><li>Given a locally constant sheaf L {\displaystyle {\mathcal {L}}} of abelian groups on <i>X</i>, let C n ( X ; L ) {\displaystyle C^{n}(X;{\mathcal {L}})} be the group of all functions <i>f</i> which map each <a href="/facts/Singular_homology/6c2AHr85"> singular <i>n</i>-simplex</a> σ : Δ n → X {\displaystyle \sigma \colon \Delta ^{n}\to X} to a global section f ( σ ) {\displaystyle f(\sigma )} of the <a href="/facts/Inverse_image_functor/wDG4ARH0"> inverse-image sheaf</a> σ − 1 L {\displaystyle \sigma ^{-1}{\mathcal {L}}} . These groups can be made into a cochain complex with differentials constructed as in usual singular cohomology. Define H s i n g j ( X ; L ) {\displaystyle H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})} to be the cohomology of this complex.</li></ul> <ul><li>The group C n ( X ~ ) {\displaystyle C_{n}({\widetilde {X}})} of singular <i>n</i>-chains on the universal cover of <i>X</i> has an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} by <a href="/facts/Covering_space/bDRGasl7"> deck transformations</a>. Explicitly, a deck transformation γ : X ~ → X ~ {\displaystyle \gamma \colon {\widetilde {X}}\to {\widetilde {X}}} takes a singular <i>n</i>-simplex σ : Δ n → X ~ {\displaystyle \sigma \colon \Delta ^{n}\to {\widetilde {X}}} to γ ∘ σ {\displaystyle \gamma \circ \sigma } . Then, given an abelian group <i>L</i> equipped with an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} , one can form a cochain complex from the groups Hom π 1 ( X , x ) ( C n ( X ~ ) , L ) {\displaystyle \operatorname {Hom} _{\pi _{1}(X,x)}(C_{n}({\widetilde {X}}),L)} of π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} -equivariant homomorphisms as above. Define H s i n g j ( X ; L ) {\displaystyle H_{\mathrm {sing} }^{j}(X;L)} to be the cohomology of this complex.</li></ul> <p>If <i>X</i> is <a href="/facts/Paracompact_space/rjTrvF7j"> paracompact</a> and <a href="/facts/Contractible_space/REXYj8uS"> locally contractible</a>, then H j ( X , L ) ≅ H s i n g j ( X ; L ) {\displaystyle H^{j}(X,{\mathcal {L}})\cong H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})} .<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> If L {\displaystyle {\mathcal {L}}} is the local system corresponding to <i>L</i>, then there is an identification C n ( X ; L ) ≅ Hom π 1 ( X , x ) ( C n ( X ~ ) , L ) {\displaystyle C^{n}(X;{\mathcal {L}})\cong \operatorname {Hom} _{\pi _{1}(X,x)}(C_{n}({\widetilde {X}}),L)} compatible with the differentials,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> so H s i n g j ( X ; L ) ≅ H s i n g j ( X ; L ) {\displaystyle H_{\mathrm {sing} }^{j}(X;{\mathcal {L}})\cong H_{\mathrm {sing} }^{j}(X;L)} . </p> <h2 id="generalization">Generalization</h2> <p>Local systems have a mild generalization to <a href="/facts/Constructible_sheaf/RkmD0R9H"> constructible sheaves</a> -- a constructible sheaf on a locally path connected topological space X {\displaystyle X} is a sheaf L {\displaystyle {\mathcal {L}}} such that there exists a stratification of </p> X = ∐ X λ {\displaystyle X=\coprod X_{\lambda }} <p>where L | X λ {\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 f : X → Y {\displaystyle f:X\to Y} . For example, if we look at the complex points of the morphism </p> f : X = Proj ( C [ s , t ] [ x , y , z ] ( s t ⋅ h ( x , y , z ) ) ) → Spec ( C [ s , t ] ) {\displaystyle f:X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x,y,z]}{(st\cdot h(x,y,z))}}\right)\to {\text{Spec}}(\mathbb {C} [s,t])} <p>then the fibers over </p> A s , t 2 − V ( s t ) {\displaystyle \mathbb {A} _{s,t}^{2}-\mathbb {V} (st)} <p>are the plane curve given by h {\displaystyle h} , but the fibers over V = V ( s t ) {\displaystyle \mathbb {V} =\mathbb {V} (st)} are P 2 {\displaystyle \mathbb {P} ^{2}} . If we take the derived pushforward R f ! ( Q _ X ) {\displaystyle \mathbf {R} f_{!}({\underline {\mathbb {Q} }}_{X})} then we get a constructible sheaf. Over V {\displaystyle \mathbb {V} } we have the local systems </p> R 0 f ! ( Q _ X ) | V ( s t ) = Q _ V ( s t ) R 2 f ! ( Q _ X ) | V ( s t ) = Q _ V ( s t ) R 4 f ! ( Q _ X ) | V ( s t ) = Q _ V ( s t ) R k f ! ( Q _ X ) | V ( s t ) = 0 _ V ( s t ) otherwise {\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}}} <p>while over A s , t 2 − V ( s t ) {\displaystyle \mathbb {A} _{s,t}^{2}-\mathbb {V} (st)} we have the local systems </p> R 0 f ! ( Q _ X ) | A s , t 2 − V ( s t ) = Q _ A s , t 2 − V ( s t ) R 1 f ! ( Q _ X ) | A s , t 2 − V ( s t ) = Q _ A s , t 2 − V ( s t ) ⊕ 2 g R 2 f ! ( Q _ X ) | A s , t 2 − V ( s t ) = Q _ A s , t 2 − V ( s t ) R k f ! ( Q _ X ) | A s , t 2 − V ( s t ) = 0 _ A s , t 2 − V ( s t ) otherwise {\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}}} <p>where g {\displaystyle g} is the genus of the plane curve (which is g = ( deg ( f ) − 1 ) ( deg ( f ) − 2 ) / 2 {\displaystyle g=(\deg(f)-1)(\deg(f)-2)/2} ). </p> <h2 id="applications">Applications</h2> <p>The cohomology with local coefficients in the module corresponding to the <a href="/facts/Orientation_covering/1SEQM2Pt">orientation covering</a> can be used to formulate <a href="/facts/Poincar%C3%A9_duality/lwNjUCIa">Poincaré duality</a> for non-orientable manifolds: see <a href="/facts/Twisted_Poincar%C3%A9_duality/XhvLatq7">Twisted Poincaré duality</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Leray_spectral_sequence/9YkJ3KWb">Leray spectral sequence</a></li> <li><a href="/facts/Gauss%E2%80%93Manin_connection/wy50u69z">Gauss–Manin connection</a></li> <li><a href="/facts/D-module/l1y6evYe">D-module</a></li> <li><a href="/facts/Intersection_homology/XnBz8gmT">Intersection homology</a></li> <li><a href="/facts/Perverse_sheaf/HyvX2c0Q">Perverse sheaf</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://math.stackexchange.com/q/13332">"What local system really is"</a>. <a href="/facts/Stack_Exchange/VQvJKJMh">Stack Exchange</a>.</li> <li>Schnell, Christian. <a href="https://www.math.stonybrook.edu/~cschnell/pdf/notes/locsys.pdf">"Computing Cohomology of Local Systems"</a> (PDF). Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex.</li> <li><a href="/facts/Geordie_Williamson/SpAmAawJ">Williamson, Geordie</a>. <a href="http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf">"An illustrated guide to perverse sheaves"</a> (PDF).</li> <li><a href="/facts/Robert_MacPherson_(mathematician)/14uI3tYv">MacPherson, Robert</a> (December 15, 1990). <a href="http://faculty.tcu.edu/gfriedman/notes/ih.pdf">"Intersection homology and perverse sheaves"</a> (PDF).</li> <li>El Zein, Fouad; Snoussi, Jawad. <a href="https://webusers.imj-prg.fr/~fouad.elzein/elzein-snoussif.pdf">"Local systems and constructible sheaves"</a> (PDF).</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Steenrod, Norman E. (1943). "Homology with local coefficients". Annals of Mathematics. 44 (4): 610–627. doi:10.2307/1969099. JSTOR 1969099. MR 0009114. <a href="/wiki/Norman_Steenrod" target="_blank">/wiki/Norman_Steenrod</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Milne, James S. (2017). Introduction to Shimura Varieties. Proposition 14.7. <a href="/wiki/James_Milne_(mathematician)" target="_blank">/wiki/James_Milne_(mathematician)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Bredon, Glen E. (1997). Sheaf Theory, Second Edition, Graduate Texts in Mathematics, vol. 25, Springer-Verlag. Chapter III, Theorem 1.1. <a href="/wiki/Glen_Bredon" target="_blank">/wiki/Glen_Bredon</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hatcher, Allen (2001). Algebraic Topology, Cambridge University Press. Section 3.H. <a href="/wiki/Allen_Hatcher" target="_blank">/wiki/Allen_Hatcher</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>