Combinatorial Algebraic Topology And Its Applications To .
Combinatorial Algebraic Topology and its Applicationsto Permutation PatternsJason P SmithUniversity of StrathclydeJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .1 / 21
Overview1Introduction to Combinatorial Algebraic TopologyBasic TopologyGraphs to Simplicial ComplexesPosets to Simplicial Complexes2Permutation PatternsIntroduction and MotivationApplying Combinatorial Algebraic TopologyKozlov, Dimitry. Combinatorial algebraic topology. Vol. 21. SpringerScience & Business Media, 2008.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .2 / 21
Simplicial ComplexesAn abstract simplicial complex is a set of subsets of some S satisfying:X and Y X Y .Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .3 / 21
Simplicial ComplexesAn abstract simplicial complex is a set of subsets of some S satisfying:X and Y X Y .S a, b, c, d, e, f , g and {{a, b, c, d}, {c, d, e}, {e, f , g }Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .3 / 21
Simplicial ComplexesAn abstract simplicial complex is a set of subsets of some S satisfying:X and Y X Y .S a, b, c, d, e, f , g and {{a, b, c, d}, {c, d, e}, {e, f , g }, {a, b, c}, {a, b, d}, {b, c, d},{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {c, e}, {d, e}, {e, f },{e, g }, {f , g }, {a}, {b}, {c}, {d}, {e}, {f }, {g }, }Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .3 / 21
Simplicial ComplexesAn abstract simplicial complex is a set of subsets of some S satisfying:X and Y X Y .S a, b, c, d, e, f , g and {{a, b, c, d}, {c, d, e}, {e, f , g }, {a, b, c}, {a, b, d}, {b, c, d},{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {c, e}, {d, e}, {e, f },{e, g }, {f , g }, {a}, {b}, {c}, {d}, {e}, {f }, {g }, }daecfgbJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .3 / 21
Simplicial ComplexesAn abstract simplicial complex is a set of subsets of some S satisfying:X and Y X Y .S a, b, c, d, e, f , g and {{a, b, c, d}, {c, d, e}, {e, f , g }, {a, b, c}, {a, b, d}, {b, c, d},{a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {c, e}, {d, e}, {e, f },{e, g }, {f , g }, {a}, {b}, {c}, {d}, {e}, {f }, {g }, }daecfgbdim 3 and non-pureJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .3 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .4 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .4 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .4 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.'Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .4 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.'Contractible homotopy equivalent to a point:Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .'4 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.'Contractible homotopy equivalent to a point:'The i’th (reduced) Betti number β̃i ( ) is the numberPof i-dimensional i”holes” and (reduced) Euler characteristic is χ̃( ) dimi 1 ( 1) βi ( )Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .4 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.'Contractible homotopy equivalent to a point:'The i’th (reduced) Betti number β̃i ( ) is the numberPof i-dimensional i”holes” and (reduced) Euler characteristic is χ̃( ) dimi 1 ( 1) βi ( )dhollowaefbcJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .4 / 21
Homotopy and HomologyTwo complexes are homotopy equivalent if we can ”continuously deform”one into the other.'Contractible homotopy equivalent to a point:'The i’th (reduced) Betti number β̃i ( ) is the numberPof i-dimensional i”holes” and (reduced) Euler characteristic is χ̃( ) dimi 1 ( 1) βi ( )dhollowβ̃ 1 0aeβ̃0 0fχ̃ 1β̃1 2bcJason P Smith (University of Strathclyde)β̃2 1Combinatorial Algebraic Topology. . .4 / 21
Graphs and the Colouring ProblembdafcJason P Smith (University of Strathclyde)eCombinatorial Algebraic Topology. . .5 / 21
Graphs and the Colouring ProblembdafceGiven a graph G how many colours do we need to colour the vertices ofthe graph so that no edge connects to two vertices of the same colour?Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .5 / 21
Graphs and the Colouring ProblembdafceGiven a graph G how many colours do we need to colour the vertices ofthe graph so that no edge connects to two vertices of the same colour?Chromatic number χ(G ) 3Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .5 / 21
Graphs to Simplicial ComplexesbdafcJason P Smith (University of Strathclyde)eCombinatorial Algebraic Topology. . .6 / 21
Graphs to Simplicial ComplexesbdafceFlag/Clique Complex Cl(G)bdafceJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .6 / 21
Graphs to Simplicial ComplexesbdafcFlag/Clique Complex Cl(G)bdaeNeighbourhood Complex N (G )ffcceadJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .be6 / 21
Other Complexes and ApplicationsJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Other Complexes and ApplicationsBipartite Complex Bip(G) ' N (G)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Other Complexes and ApplicationsBipartite Complex Bip(G) ' N (G)Independence Complex Ind(G) ' Cl(Ḡ)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Other Complexes and ApplicationsBipartite Complex Bip(G) ' N (G)Independence Complex Ind(G) ' Cl(Ḡ)Colouring ComplexJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Other Complexes and ApplicationsBipartite Complex Bip(G) ' N (G)Independence Complex Ind(G) ' Cl(Ḡ)Colouring ComplexHom Complex Hom(G, H)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Other Complexes and ApplicationsBipartite Complex Bip(G) ' N (G)Independence Complex Ind(G) ' Cl(Ḡ)Colouring ComplexHom Complex Hom(G, H)Lovász Complex Lo(G ) : (N(F(N (G ))))Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Other Complexes and ApplicationsBipartite Complex Bip(G) ' N (G)Independence Complex Ind(G) ' Cl(Ḡ)Colouring ComplexHom Complex Hom(G, H)Lovász Complex Lo(G ) : (N(F(N (G ))))Graphs in Metric Spaces such as Rips Complex, Alpha Complex,Witness Complex . . .Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Other Complexes and ApplicationsBipartite Complex Bip(G) ' N (G)Independence Complex Ind(G) ' Cl(Ḡ)Colouring ComplexHom Complex Hom(G, H)Lovász Complex Lo(G ) : (N(F(N (G ))))Graphs in Metric Spaces such as Rips Complex, Alpha Complex,Witness Complex . . .More . . .Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .7 / 21
Posets to Simplicial ComplexesgchedafbJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .8 / 21
Posets to Simplicial ComplexesgchedafChains of a poset are the totallyordered subsets.E.g. {a c g }bJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .8 / 21
Posets to Simplicial ComplexesgchedafChains of a poset are the totallyordered subsets.E.g. {a c g } and {b h}.bJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .8 / 21
Posets to Simplicial ComplexesgchedadbbgcfChains of a poset are the totallyordered subsets.E.g. {a c g } and {b h}.fhaChains of a poset P give faces of theOrder Complex (P).eJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .8 / 21
Posets to Simplicial ComplexesgchedadbbgcfChains of a poset are the totallyordered subsets.E.g. {a c g } and {b h}.fhaChains of a poset P give faces of theOrder Complex (P).eJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .8 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂.P Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂.1̂0̂Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂.1Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂. 1 1 11Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂.2 11 1 11Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂. 12 11 1 11Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
Möbius functionThe Möbius function for a poset is defined as µ(a, b) 0 if a 6 b,µ(a, a) 1 for all a and for a b:Xµ(a, b) µ(a, z).a z bTo calculate µ(P) add a top and bottom element 1̂ and 0̂. 121µ(P) 1 1 1 11Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .9 / 21
ApplicationsLemmaµ(P) χ̃( (P))Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (P) ' (Q) µ(P) µ(Q)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (P) ' (Q) µ(P) µ(Q)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (P) ' (Q) µ(P) µ(Q) Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (P) ' (Q) µ(P) µ(Q) 'Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (P) ' (Q) µ(P) µ(Q) 'Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (P) ' (Q) µ(P) µ(Q)µ 1 'Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (P) ' (Q) µ(P) µ(Q)µ 1µ 1 'Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .10 / 21
Applications (Z ) (P) \ (Q) µ(Z ) µ(P) µ(Q)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .11 / 21
Applications (Z ) (P) \ (Q) µ(Z ) µ(P) µ(Q) (P) susp( (Q)) µ(P) µ(Q)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .11 / 21
Applications (Z ) (P) \ (Q) µ(Z ) µ(P) µ(Q) (P) susp( (Q)) µ(P) µ(Q) (Z ) (P) (Q) µ(Z ) µ(P) µ(Q) µ(P Q)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .11 / 21
Applications (Z ) (P) \ (Q) µ(Z ) µ(P) µ(Q) (P) susp( (Q)) µ(P) µ(Q) (Z ) (P) (Q) µ(Z ) µ(P) µ(Q) µ(P Q) (Z ) (P) (Q) µ(Z ) µ(P)µ(Q)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .11 / 21
Applications (Z ) (P) \ (Q) µ(Z ) µ(P) µ(Q) (P) susp( (Q)) µ(P) µ(Q) (Z ) (P) (Q) µ(Z ) µ(P) µ(Q) µ(P Q) (Z ) (P) (Q) µ(Z ) µ(P)µ(Q) (Z ) (P) ? (Q) µ(Z ) µ(P)µ(Q)Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .11 / 21
A Nice Counting Application of the Möbius FunctionLet [n] : {1, . . . , n} and X k : {x [n] x k} computeX( 1) A A X k A [n]Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .12 / 21
A Nice Counting Application of the Möbius FunctionLet [n] : {1, . . . , n} and X k : {x [n] x k} computeX( 1) A A X k A [n]Proposition (Crosscut Theorem)Consider poset P and subset X s.t p P x X s.t p x, then:X( 1) A .µ(0̂, 1̂) A X A 1̂Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .12 / 21
A Nice Counting Application of the Möbius FunctionLet [n] : {1, . . . , n} and X k : {x [n] x k} computeX( 1) A A X k A [n]{1, 2, 3, 4}B4{1, 2, 3}{1, 2}{1, 2, 4}{1, 3}{1, 3, 4}{2, 3}{1}{1, 4}{2}{3}{2, 3, 4}{2, 4}{3, 4}{4} µ(Bn ) ( 1)nJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .12 / 21
A Nice Counting Application of the Möbius FunctionLet [n] : {1, . . . , n} and X k : {x [n] x k} computeX( 1) A A X k A [n]{1, 2, 3, 4}B4 2{1, 2, 3}{1, 2}{1, 2, 4}{1, 3}{1, 3, 4}{2, 3}{1, 4}{2, 3, 4}{2, 4}{3, 4}0̂µ(Bn ) ( 1)nJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .12 / 21
A Nice Counting Application of the Möbius FunctionLet [n] : {1, . . . , n} and X k : {x [n] x k} computeX( 1) A A X k A [n]{1, 2, 3, 4}B4 2{1, 2, 3}{1, 2}{1, 2, 4}{1, 3}{1, 3, 4}{2, 3}{1, 4}{2, 3, 4}{2, 4}0̂µ(Bn ) ( 1)nJason P Smith (University of Strathclyde)andµ(Bn k ) ( 1)n k 1Combinatorial Algebraic Topology. . . n 1k 1{3, 4} 12 / 21
A Nice Counting Application of the Möbius FunctionLet [n] : {1, . . . , n} and X k : {x [n] x k} computeX( 1) A A X k A [n]{1, 2, 3, 4}B4 2{1, 2, 3}{1, 2}{1, 2, 4}{1, 3}{1, 3, 4}{2, 3}{1, 4}{2, 3, 4}{2, 4}0̂µ(Bn ) ( 1)nXandµ(Bn k ) A ( 1) n k 1 ( 1)A X k A [n]Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .n 1k 1 n 1 k 1( 1)n k 1{3, 4} 12 / 21
Permutation PatternsSingle line notation for permutations i.e 241365.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .13 / 21
Permutation PatternsSingle line notation for permutations i.e 241365.An occurrence of σ in π is a subsequence of π with the same relative orderof size as the letters in σe.g. 132 occurs twice in 23541.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .13 / 21
Permutation PatternsSingle line notation for permutations i.e 241365.An occurrence of σ in π is a subsequence of π with the same relative orderof size as the letters in σe.g. 132 occurs twice in 23541.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .13 / 21
Permutation PatternsSingle line notation for permutations i.e 241365.An occurrence of σ in π is a subsequence of π with the same relative orderof size as the letters in σe.g. 132 occurs twice in 23541.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .13 / 21
Permutation PatternsSingle line notation for permutations i.e 241365.An occurrence of σ in π is a subsequence of π with the same relative orderof size as the letters in σe.g. 132 occurs twice in 23541.Permutation poset P contains all permutations and σ π if σ occurs in π.An interval of P is [σ, π] {τ σ τ π}, rank π σ .Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .13 / 21
Permutation PatternsSingle line notation for permutations i.e 241365.An occurrence of σ in π is a subsequence of π with the same relative orderof size as the letters in σe.g. 132 occurs twice in 23541.Permutation poset P contains all permutations and σ π if σ occurs in π.An interval of P is [σ, π] {τ σ τ π}, rank π σ .In 1968 Donald Knuth showed the permutations that can be sorted by astack are the permutations that avoid 231Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .13 / 21
Permutation PatternsSingle line notation for permutations i.e 241365.An occurrence of σ in π is a subsequence of π with the same relative orderof size as the letters in σe.g. 132 occurs twice in 23541.Permutation poset P contains all permutations and σ π if σ occurs in π.An interval of P is [σ, π] {τ σ τ π}, rank π σ .In 1968 Donald Knuth showed the permutations that can be sorted by astack are the permutations that avoid 231Lots of work in enumerating avoidance of permutations. Studying theMöbius function and topology of P can help with this.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .13 / 21
The Interval [123, 1234123Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .14 / 21
The Interval [123, n P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .14 / 21
The Interval [123, �̃ 0234123451Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .14 / 21
The Interval [123, 1234451231233451234561245612341231234µ 0contractibleχ̃ 0234123451Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .14 / 21
Previous ResultsJason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .15 / 21
Previous ResultsThe direct sum is σ π σ1 . . . σm (π1 m) . . . (πn m), for example,312 213 312546.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .15 / 21
Previous ResultsThe direct sum is σ π σ1 . . . σm (π1 m) . . . (πn m), for example,312 213 312546.(Sagan and Vatter, 2006): Intervals of layered permutations, that is,permutations that are the direct sum of two or more decreasingpermutations.(Björner, Jelinek, Jelinek and Steı́ngrimsson, 2011): Intervals ofseparable permutations, that is, permutations that avoid 2413 and3142.(Björner et al., 2011): Intervals of decomposable permutations, thatis, permutations that can be written as the direct sum of two or morenon-empty permutations.Jason P Smith (University of Strathclyde)Combinatorial Algebraic Topology. . .15 / 21
Previous ResultsThe direct sum is σ π σ1 . . . σm (π1 m) . . . (πn m), for example,312 213 312546.(Sagan and Vatter,
1 Introduction to Combinatorial Algebraic Topology Basic Topology Graphs to Simplicial Complexes Posets to Simplicial Complexes 2 Permutation Patterns Introduction and Motivation Applying Combinatorial Algebraic Topology Kozlov, Dimitry. Combinatorial algebraic topology. Vol. 21. Springer Science & Business Media, 2008.
in combinatorial topology he needed to prove, which has no counterpart in algebraic topology. Even if the backbone of this course is combinatorial topology, the applications in combina- torics will play a centrale role, for they ultimately remain the true motivation.
Combinatorial Algebraic Topology With 115 Figures and 1 Table fyj Springer. Contents Overture Part I Concepts of Algebraic Topology 2 Cell Complexes 7 2 1 Abstract Simphcial Complexes 7 2 11 Definition of Abstract Simphcial Complexes and Maps Between Them 7 2 12 Deletion, Link, Star, and Wedge 10
Zvi Rosen Applied Algebraic Topology Notes Vladimir Itskov 1. August 24, 2015 Algebraic topology: take \topology" and get rid of it using combinatorics and algebra. Topological space 7!combinatorial object 7!algebra (a bunch of vector spaces with maps).
topology and di erential geometry to the study of combinatorial spaces. Per-haps surprisingly, many of the standard ingredients of di erential topology and di erential geometry have combinatorial analogues. The combinatorial theories This work was partially supported by the National Science Foundation and the National Se-curity Agency. 177
of points with the entire collection often regarded as a space; and combinatorial or algebraic topology, which treats geometrical figures as aggregates of smaller building blocks, just as a wall is a collection of bricks. 1 Of course notions of point set topology are used in combinatorial topology, especially for very general geometric .
Concurrency and directed algebraic topology Martin Raussen Department of Mathematical Sciences, Aalborg University, Denmark Applied and Computational Algebraic Topology 6ECM July 4, 2012 . A combinatorial model and its geometric realization Combinatorics poset category C(X)(0,1) MR,
Any topology over [0;1](S) de nes a topology over jKj, but combinatorial topol-ogy most often is not concerned by such a topology: the combinatorial game is enough to model, up to homotopy, in this way most \sensible" topological spaces. 2.2 Simple examples. Let V be an arbitrary set of vertices, possibly in nite. Then the simplex generated
Keywords: Korean, heritage language, multiliteracies, university-level language classroom, multimodal reading response Journal of Language and Literacy Education Vol. 11 Issue 2—Fall 2015 117 eritage language (HL) learners1 who are exposed to and speak a language other than English exclusively in their homes and communities exhibit relatively lower reading and writing skills compared to .
