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If one fixes a  basepoint, there is a canonical way of embedding the lower-dimensional symmetric products into the higher-dimensional ones. That way, one can consider the colimit over the symmetric products, the infinite symmetric product. This construction can easily be extended to give a homotopy functor.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4From an algebraic point of view, the infinite symmetric product is the free commutative monoid generated by the space minus the basepoint, the basepoint yielding the identity element. That way, one can view it as the abelian version of the James reduced product. One of its essential applications is the Dold-Thom theorem, stating that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as the reduced homology groups of that complex. That way, one can give a homotopical definition of  homology.         (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsDefinition [ edit ] Examples [ edit ] Properties [ edit ] H-space structure [ edit ] Functioriality [ edit ] Simplicial and CW structure [ edit ] Homotopy [ edit ] Homology [ edit ] Related constructions and generalisations [ edit ] References [ edit ] External Links [ edit ] Definition [ edit ] Let X be a topological space and n \u2265 1 a natural number. Define the n th symmetric product of X or the n -fold symmetric product of X as the space SP n\u2061 ( X ) = X n\/ S n. {Displaystyle OperatorName {sp} ^{n} (x) = x ^{n}\/s_ {n}.} Here, the symmetric group S n acts on X n by permuting the factors. Hence, the elements of SP n ( X ) are the unordered n -tuples of elements of X . Write [ x first , …, x n ] for the point in SP n ( X ) defined by ( x first , …, x n ) \u2208 X n .        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Note that one can define the n th symmetric product in any  category where  products and colimits exist. Namely, one then has canonical  isomorphisms \u03c6\u00a0: X \u00d7 AND \u2192 AND \u00d7 X for any objects X and AND and can define the  action of the transposition ( k  k + first ) \u2208 S n {displaystyle (k k+1)in S_{n}} on X n as Id k – first \u00d7 \u03d5 \u00d7 Id n – k – first {displaystyle operatorname {Id} ^{k-1}times phi times operatorname {Id} ^{n-k-1}} , thereby inducing an action of the whole S n on X n . This means that one can consider symmetric products of objects like  simplicial sets as well. Moreover, if the category is cartesian closed, the distributive law X \u00d7 ( AND \u2210 WITH ) \u2245 X \u00d7 AND \u2210 X \u00d7 WITH holds and therefore one gets SP n\u2061 ( X \u2a3f AND ) = \u2210 k=0nSP k\u2061 ( X ) \u00d7 SP n\u2212k\u2061 ( AND ) . {displaystyle operatorname {SP} ^{n}(Xamalg Y)=coprod _{k=0}^{n}operatorname {SP} ^{k}(X)times operatorname {SP} ^{n-k}(Y).} If ( X , It is ) is a based space, it is common to set SP 0 ( X ) = { It is }. Further, X n can then be embedded into X n +1 by sending ( x first , …, x n ) to ( x first , …, x n , It is ). This clearly induces an embedding of SP n ( X ) into SP n +1 ( X ). Therefore, the infinite symmetric product can be defined as SP \u2061 ( X ) = colim \u2061 SP n\u2061 ( X ) . {displaystyle operatorname {SP} (X)=operatorname {colim} operatorname {SP} ^{n}(X).} A definition avoiding category theoretic notions can be given by taking SP( X ) to be the union of the increasing sequence of spaces SP n ( X ) equipped with the direct limit topology. This means that a subset of SP( X ) is open if and only if all its intersections with the SP n ( X ) are open. We define the basepoint of SP( X ) as [ It is ]. That way, SP( X ) becomes a based space as well. One can generalise this definition as well to  pointed categories where products and colimits exist. Namely, in this case one has a canonical map X n \u2192 X n +1 , induced by the identity X n \u2192 X n and the zero map X n \u2192 X . So this results in a direct system of the symmetric products, too and one can therefore define its colimit as the infinite symmetric product. Examples [ edit ] SP n ( I ) is the same as the n -dimensional standard simplex \u0394 n , where I denotes the unit interval. SP n ( S first ) can be identified with the space of conjugacy classes of  unitary n \u00d7 n -matrices, where S first is supposed to be the circle. This is because such a class is uniquely determined by the eigenvalues of an element of the class, all lying in S first . At first, one can easily see that this space is homotopy-equivalent to S first : As SP n is a homotopy functor (see  Properties), the space in question is homotopy-equivalent to SP n ( C \u2212 {0}). Consider the map SP n ( C – {0}) \u2192 P n into the space P n of polynomials over C of degree at most n , mapping [ In first , …, In n ] to ( With – In first ) \u22c5\u22c5\u22c5 ( With – In n ). This way, one can identify SP n ( C \u2212 {0}) with the space of monic polynomials of degree n having constant term different from zero, i.e. C n – 1 \u00d7 ( C \u2212 {0}), which is homotopy-equivalent to S first . This implies that the infinite symmetric product SP( S first ) is homotopy-equivalent to S first as well. However, one knows considerably more about the space SP n ( S first ). Namely, that the map SPn\u2061(S1)\u2192S1,.[w1,\u2026,wn]\u21a6w1\u22efwn{displaystyle {begin{aligned}qquad operatorname {SP} ^{n}(S^{1})&to S^{1},\\{color {white}.}[w_{1},dots ,w_{n}]&mapsto w_{1}cdots w_{n}end{aligned}}} is a fibre bundle with fibre being homeomorphic to the ( n \u2212 1)-dimensional standard simplex \u2206 n \u22121 . It is orientable if and only if n is odd. [first] [2] SP( S 2 ) is homeomorphic to the infinite-dimensional complex projective space CP \u221e as follows: The space CP n can be identified with the space of nonzero polynomials of degree at most n over C up to scalar multiplication by sending a 0 + … + a n With n to the line passing through ( a 0 , …, a n ). Interpreting S 2 as the Riemann sphere C \u222a {\u221e} yields a map f:(S2)n\u2192CPn,(a1,\u2026,an)\u21a6(z+a1)\u22ef(z+an),{displaystyle {begin{aligned}qquad fcolon (S^{2})^{n}&to mathbf {CP} ^{n},\\(a_{1},dots ,a_{n})&mapsto (z+a_{1})cdots (z+a_{n}),end{aligned}}} where the possible factors With + \u221e are omitted. One can check that this map indeed is continuous. [3] As f ( a first , …, a n ) remains unchanged under permutation of the a i ‘s, f induces a continuous bijection SP n (S 2 ) \u2192 CP n . But as both are  compact Hausdorff spaces, this map is a homeomorphism. Letting n go to infinity shows that the assertion holds.  Although calculating SP( S n ) for n \u2265 3 turns out to be quite difficult, one can still describe SP 2 ( S n ) quite well as the mapping cone of a map \u03a3 n Rp n-1 \u2192 S n , where s n stands for applying the reduced suspension n times and Rp n \u22121 is the ( n \u2212 1)-dimensional real projective space: One can view SP 2 ( S n ) as a certain quotient of D n \u00d7 D n by identifying S n with D n \/\u2202 D n . Interpreting D n \u00d7 D n as the cone on its boundary D n \u00d7 \u2202 D n \u222a \u2202 D n \u00d7 D n , the identifications for SP 2 respect the concentric copies of the boundary. Hence, it suffices to only consider these. The identifications on the boundary \u2202 D n \u00d7 D n \u222a D n \u00d7 \u2202 D n of D n \u00d7 D n itself yield S n . This is clear as this is a quotient of D n \u00d7 \u2202 D n and as \u2202 D n is collapsed to one point in S n .The identifications on the other concentric copies of the boundary yield the quotient space WITH of D n \u00d7 \u2202 D n , obtained by identifying ( x , and ) with ( and , x ) whenever both coordinates lie in \u2202 D n . Define a map f : D n \u00d7 Rp n \u22121 \u2192 WITH by sending a pair ( x , L ) to ( In , With ). Here, With \u2208 \u2202 D n and In \u2208 D n are chosen on the line through x parallel to L such that x is their midpoint. If x is the midpoint of the segment zz\u2032 , there is no way to distinguish between With and In , but this is not a problem since f takes values in the quotient space WITH . Therefore, f is well-defined. As f ( x , L ) = f ( x , L\u2032 ) holds for every x \u2208 \u2202 D n , f factors through \u03a3 n Rp n \u22121 and is easily seen to be a homeomorphism on this domain. Properties [ edit ] H-space structure [ edit ] As SP( X ) is the free commutative monoid generated by X – { It is } with identity element It is , it can be thought of as a commutative analogue of the James reduced product J ( X ). This means that SP( X ) is the quotient of J ( X ) obtained by identifying points that differ only by a permutation of coordinates. Therefore, the H-space structure on J ( X ) induces one on SP( X ) if X is a CW complex, making it a commutative and associative H-space with strict identity. As such, the Dold-Thom theorem implies that all its k -invariants vanish, meaning that it has the  weak homotopy type of a generalised Eilenberg-MacLane space if X is path-connected. [4] However, if X is an arbitrary space, the multiplication on SP( X ) may fail to be continuous. [5] Functioriality [ edit ] SP n is a homotopy functor: A map f : X \u2192 AND clearly induces a map SP n ( f )\u00a0: SP n ( X ) \u2192 SP n ( AND ) given by SP n ( f ) [ x first , …, x n ] = [ f ( x first ), …, f ( x n )]. A homotopy between two maps f , g : X \u2192 AND yields one between SP n ( f ) and SP n ( g ). Also, one can easily see that the diagram commutes, meaning that SP is a functor as well. Similarly, SP is even a homotopy functor on the category of pointed spaces and basepoint-preserving homotopy classes of maps. In particular, X \u2243 AND implies SP n ( X ) \u2243 SP n ( AND ), but in general not SP( X ) \u2243 SP( AND ) as homotopy equivalence may be affected by requiring maps and homotopies to be basepoint-preserving. However, this is not the case if one requires X and AND to be connected CW complexes. [6] Simplicial and CW structure [ edit ] SP( X ) inherits certain structures of X : For a simplicial complex X , one can also install a simplicial structure on X n such that each n -permutation is either the identity on a simplex or a homeomorphism from one simplex to another. This means that one gets a simplicial structure on SP n ( X ). Furthermore, SP n ( X ) is also a subsimplex of SP n +1 ( X ) if the basepoint It is \u2208 X is a vertex, meaning that SP( X ) inherits a simplicial structure in this case as well. [7] However, one should note that X n and SP n ( X ) do not need to have the weak topology if X has uncountably many simplices. [8] An analogous statement can be made if X is a CW complex. Nevertheless, it is still possible to equip SP( X ) with the structure of a CW complex such that both topologies have the same compact sets if X is an arbitrary simplicial complex. [9] So the distinction between the two topologies will not cause any differences for purposes of homotopy, e.g. Homotopy [ edit ] One of the main uses of infinite symmetric products is the Dold-Thom theorem. It states that the reduced homology groups coincide with the homotopy groups of the infinite symmetric product of a connected CW complex. This allows one to reformulate homology only using homotopy which can be very helpful in algebraic geometry. It also means that the functor SP maps  Moore spaces M ( G , n ) to Eilenberg-MacLane spaces K ( G , n ). Therefore, it yields a natural way to construct the latter spaces given the proper Moore spaces. It has also been studied how other constructions combined with the infinite symmetric product affect the homotopy groups. For example, it has been shown that the map r : SP \u2061 ( X ) \u2192 Oh SP \u2061 ( A X ) , r [ x 1, … , x n] ( t ) = [ ( x 1, t ) , … , ( x n, t ) ] {displaystyle rho colon operatorname {SP} (X)to Omega operatorname {SP} (Sigma X),quad rho [x_{1},dots ,x_{n}](t)=[(x_{1},t),dots ,(x_{n},t)]} is a weak homotopy equivalence, where \u03a3 X = X \u2227 S first denotes the reduced suspension and \u03a9 AND stands for the loop space of the pointed space AND . [ten] Homology [ edit ] Unsurprisingly, the homology groups of the symmetric product cannot be described as easily as the homotopy groups. Nevertheless, it is known that the homology groups of the symmetric product of a CW complex are determined by the homology groups of the complex. More precisely, if X and AND are CW complexes and R is a principal ideal domain such that H i ( X , R ) \u2245 H i ( AND , R ) for all i \u2264 k , then H i (SP n ( X ), R ) \u2245 H i (SP n ( AND ), R ) holds as well for all i \u2264 k . This can be generalised to \u0393-products, defined in the next section. [11] For a simplicial set K , one has furthermore H \u2217( SP n+1\u2061 ( K ) ) \u2245 H \u2217( SP n+1\u2061 ( K ) , SP n\u2061 ( K ) ) \u2295 H \u2217( SP n\u2061 ( K ) ) . {displaystyle H_{*}(operatorname {SP} ^{n+1}(K))cong H_{*}(operatorname {SP} ^{n+1}(K),operatorname {SP} ^{n}(K))oplus H_{*}(operatorname {SP} ^{n}(K)).} Passing to geometric realisations, one sees that this statement holds for connected CW complexes as well. [twelfth] Induction yields furthermore H \u2217( SP \u2061 ( K ) ) \u2245 \u2a01 n=1\u221eH \u2217( SP n\u2061 ( K ) , SP n\u22121\u2061 ( K ) ) . {displaystyle H_{*}(operatorname {SP} (K))cong bigoplus _{n=1}^{infty }H_{*}(operatorname {SP} ^{n}(K),operatorname {SP} ^{n-1}(K)).} [13] Related constructions and generalisations [ edit ] S. Liao introduced a slightly more general version of symmetric products, called \u0393-products for a subgroup \u0393 of the symmetric group S n . [14] The operation was the same and hence he defined X C = X n \/\u0393 as the \u0393-product of X . That allowed him to study cyclic products , the special case for \u0393 being the cyclic group, as well. When establishing the Dold-Thom theorem, they also considered the “quotient group” WITH [ X ] of SP( X ). This is the free abelian group over X with the basepoint as the zero element. If X is a CW complex, it is even  a topological group. In order to equip this group with a topology,  Dold and  Thom initially introduced it as the following quotient over the infinite symmetric product of the wedge sum of X with a copy of itself: Let \u03c4\u00a0: X \u2228 X \u2192 X \u2228 X be interchanging the summands. Furthermore, let ~ be the equivalence relation on SP( X \u2228 X ) generated by x \u223c x + and + SP \u2061 ( T ) ( and ) {displaystyle xsim x+y+operatorname {SP} (tau )(y)} for x , and \u2208 SP( X \u2228 X ). Then one can define WITH [ X ] as WITH [ X ] = SP \u2061 ( X \u2228 X ) \/ \u223c . {displaystyle mathbb {Z} [X]=operatorname {SP} (Xvee X)\/sim .} Since ~ is compatible with the addition in SP( X \u2228 X ), one gets an associative and commutative addition on WITH [ X ]. One also has the topological inclusions X \u2282 SP( X ) \u2282 WITH [ X ] [15] and it can easily be seen that this construction has properties similar to the ones of SP, like being a functor. McCord gave a construction generalising both SP( X ) and WITH [ X ]: Let G be a monoid with identity element 1 and let ( X , It is ) be a pointed set. Define B ( G , X ) = { in : X \u2192 G : in ( It is ) = first \u00a0and\u00a0 in ( x ) = first \u00a0for all but finitely many\u00a0 x \u2208 X } . {displaystyle B(G,X)={ucolon Xto G:u(e)=1{text{ and }}u(x)=1{text{ for all but finitely many }}xin X}.} Then B ( G , X ) is again a monoid under pointwise multiplication which will be denoted by \u22c5. Let gx denote the element of B ( G , X ) taking the value g at x and being 1 elsewhere for g \u2208 G , x \u2208 X – { It is }. Moreover, ge shall denote the function being 1 everywhere, the unit of B ( G , X ). In order to install a topology on B ( G , X ), one needs to demand that X be compactly generated and that G be an abelian topological monoid. Define B n ( G , X ) to be the subset of B ( G , X ) consisting of all maps that differ from the constant function 1 at no more than n points. B n ( G , X ) gets equipped with the final topology of the map \u03bcn:(G\u00d7X)n\u2192Bn(G,X),((g1,x1),\u2026,(gn,xn))\u21a6g1x1\u22efgnxn.{displaystyle {begin{aligned}mu _{n}colon (Gtimes X)^{n}&to B_{n}(G,X),\\((g_{1},x_{1}),dots ,(g_{n},x_{n}))&mapsto g_{1}x_{1}cdots g_{n}x_{n}.end{aligned}}} Now, B n ( G , X ) is a closed subset of B n+1 ( G , X ). [16] Then B ( G , X ) can be equipped with the direct limit topology, making it again a compactly generated space. One can then identify SP( X ) respectively WITH [ X ] with B ( N , X ) respectively B ( WITH , X ). Moreover, B (\u22c5,\u22c5) is functorial in the sense that B : C \u00d7 D \u2192 C is a bifunctor for C being the category of abelian topological monoids and D being the category of pointed CW complexes. [17] Here, the map B (Phi, f ): B ( G , X ) \u2192 B ( H , AND ) for a morphism \u03c6: G \u2192 H of abelian topological monoids and a continuous map f : X \u2192 AND is defined as B ( Phi , f ) ( g 1x 1\u22ef g nx n) = ( Phi g 1) ( f x 1) \u22ef ( Phi g n) ( f x n) {displaystyle B(varphi ,f)(g_{1}x_{1}cdots g_{n}x_{n})=(varphi g_{1})(fx_{1})cdots (varphi g_{n})( fx_ {n})} for all g i \u2208 G and x i \u2208 X . As in the preceding cases, one sees that a based homotopy f t : X \u2192 AND induces a homotopy B (Id, f t ): B ( G , X ) \u2192 B ( G , AND ) for an abelian topological monoid G . Using this construction, the Dold-Thom theorem can be generalised. Namely, for a discrete module M over a commutative ring with unit one has [ X , B ( M , AND ) ] \u2245 \u220f n=0\u221eH~n( X , H~n( AND , M ) ) {Displaystyle [X, B (M, Y)] CONG Prod _ {n = 0}^{Infty} {tILDE {H}}^{N} (x, {tILDE {H}} _ {n} (y, M))} for based spaces X and AND having the homotopy type of a CW complex. [18] Here, H\u0303 n denotes reduced homology and [ X , WITH ] stands for the set of all based homotopy classes of basepoint-preserving maps X \u2192 WITH . As M is a module, [ X , B ( M , AND )] has an obvious group structure. Inserting X = S n and M = WITH yields the Dold-Thom theorem for WITH [ X ]. It is noteworthy as well that B ( G , S first ) is a classifying space for G if G is a topological group such that the inclusion {1} \u2192 G is a cofibration. [19] ^  Morton, Hugh R. (1967). “Symmetric Products of the Circle”. Mathematical Proceedings of the Cambridge Philosophical Society . Vol.\u00a063. Cambridge University Press. pp.\u00a0349\u2013352.  ^  Symmetric Product of Circles on the nlab  ^  Hatcher (2002), Example 4K.4  ^  Dold and Thom (1958), sentence 7.1  ^  Spaniard (1959), Footnote 2  ^  Hatcher (2002), p.481  ^  Aguilar, Gitler and Prieto (2008), Note 5.2.2  ^  Dold and Thom (1958), 3.3  ^  Hatcher (2002), pp.482-483  ^  Spaniard (1959), theorem 10.1  ^  Dold (1958), Theorem 7.2  ^  Milgram, R. James (1969), “The Homology of Symmetric Products”, Transactions of the American Mathematical Society , 138 : 251\u2013265  ^  Spaniard (1959), Theorem 7.2  ^  Liao (1954)  ^  Dold and Thom (1958), 4.7  ^  McCord (1969), Lemma 6.2  ^  McCord (1969), Corollary 6.9  ^  McCord (1969), Theorem 11.5  ^  McCord (1969), Theorem 9.17  References [ edit ] Aguilar, Marcelo; Gitler, Samuel; Prieto, Carlos (2008). Algebraic Topology from a Homotopical Viewpoint . Springer Science & Business Media. ISBN\u00a0 978-0-387-22489-3 . Dold, Albrecht (1958), “Homology of Symmetric Products and other Functor of Complexes”, Annals of Mathematics : 54\u201380 Dold, Albrecht; Thom, Ren\u00e9 (1958), “Quasifasis and Endless Symmetrical Products”, Annals of Mathematics , Second Series, sixty seven (2): 239\u2013281, doi: 10,2307\/1970005 , ISSN\u00a0 0003-486X , JSTOR\u00a0 1970005 , MR\u00a0 0097062 Hatcher, Allen (2002). Algebraic Topology . Cambridge University Press. ISBN\u00a0 978-0-521-79540-1 . Liao, S.D. (1954), “On the Topology of Cyclic Products of Spheres”, Transactions of the American Mathematical Society , 77 (3): 520\u2013551 McCord, Michael C. (1969), “Classifying Spaces and Infinite Symmetric Products”, Transactions of the American Mathematical Society , 146 : 273\u2013298 Piccinini, Renzo A. (1992). Lectures on Homotopy Theory . Elsevier. ISBN\u00a0 9780080872827 . Spanier, Edwin (1959), “Infinite Symmetric Products, Function Spaces and Duality”, Annals of Mathematics : 142\u2013198 External Links [ edit ]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/symmetric-product-topology-wikipedia\/#breadcrumbitem","name":"Symmetric product (topology) – Wikipedia"}}]}]