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The locked cohomology of the torus
Topology and its Applications 221 (2017) 156–166
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Topology and its Applications www.elsevier.com/locate/topol
Virtual Special Issue – Dedicated to the 120th anniversary of the eminent Russian mathematician P.S. Alexandroff
The locked cohomology of the torus ✩ Igor Usimov, S.M. Ageev ∗ Belarus State University, Minsk, Belarus
a r t i c l e
i n f o
Article history: Received 13 July 2016 Accepted 15 November 2016 Available online 11 February 2017
a b s t r a c t We calculate the ring of the locked cohomology of the torus. © 2017 Elsevier B.V. All rights reserved.
MSC: 55N91 55P20 55M15 55R40 Keywords: Isovariant absolute extensors Universal G-space in the sense of Palais Equivariant tom Dieck cohomologies Universal G-space in the sense of tom Dieck
1. Introduction For a semi-ideal F ⊂ ConjG of orbit types, a universal F-space EF (in the sense of tom Dieck [1]) generates an equivariant cohomology theories HF∗ (called the tom Dieck cohomology theories) which assigns the ring HF∗ (X) = H ∗ ((EF ×X)/G; Q) of the Čech cohomology to X (here Q is the field of rational numbers). A partial case of HF∗ is the equivariant Borel cohomology corresponding F = {(e)}. The tom Dieck cohomology possesses many convenient formal properties, but its computation is difficult (in comparison with the classical Borel equivariant cohomology). However, within the framework of the theory of the universal Palais G-spaces (in other words, the theory of isovariant extensors, or Isov-AE-spaces, see [2]), there appears a new possibility for construction and computation of the tom Dieck cohomology. It turns out that universal F-spaces exist for all families F of orbit types, and all of them are concentrated within an Isov-AE-space W (the so-called effect of concentration): WF is a F-universal space in the sense ✩
This paper was written with the partial support of a grant from the Ministry of Education of the Republic of Belarus.
* Corresponding author. E-mail address: [email protected] (S.M. Ageev). http://dx.doi.org/10.1016/j.topol.2017.02.033 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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of tom Dieck. Moreover, they have the properties of mutual location in W as good as it can be possible. This method allows to bring the computations of some cohomology invariants to the explicit formulae. ˇ ∗ (WF = WF /G; Q), we have a strong need for its Since HF∗ (X) is an algebra over the ring HF∗ (∗) = H calculation. In general, this problem seems to be quite difficult. Even for a compact connected Lie group G, the problem of computation of the cohomology HP∗ (∗) where P = ConjG \{(e)} (which we prefer to call a locked cohomology) is still far from being completely solved, though it presents an considerable interest in the homogeneity conjecture for the Banach–Mazur compactum (in particular, the establishment of its non-triviality) (see [3]). It is known that HP∗ (∗) is the polynomial ring Q[x] with generator x of degree 2 (for G = S 1 ) and 4 (for G = O(2)) (see [2]). The aim of the paper is the computation of the locked cohomology for G = T2 = S 1 × S 1 . ⎧ ⎪ ⎪ ⎨Q, if q = 0 q ˇ (WP , Q) = 0, if q = 1 or q = 2k, k ∈ N Theorem 1. H ⎪ ⎪ ⎩Qω , if q = 2k + 1, k ∈ N,
where Qω denotes the countable power of Q.
Theorem 1 implies that the multiplication in HP∗ (∗) is trivial. 2. Preliminary facts and results In what follows we shall assume all spaces (all maps) to be metric (continuous, respectively), if they do not arise as a result of some constructions or if the opposite is not claimed; all acting groups are assumed to be compact Lie groups. We present the basic notions of the theory of G-spaces [4]. An action of a compact group G on a space X is a continuous map μ from the product G × X into X satisfying the following properties: 1. μ(g, μ(h, x)) = μ(g · h, x); and 2. μ(e, x) = x for all x ∈ X, g, h ∈ G (here e is the unit of the group G). As a rule, μ(g, x) will be written as g · x or just gx. A space X with an action of the group G is called a G-space. The map f : X → Y of G-spaces is called a G-map or an equivariant map if f (g · x) = g · f (x) for all x ∈ X, g ∈ G. The subset {g · x | g ∈ G} = G · x is called the orbit G(x) of the point x ∈ X which turns out to be closed. The natural map π = πX : X → X/G, x → G(x), of the space X onto the space X/G of quotient partition is said to be the orbit projection. We call the space of quotient partition, equipped with the quotient topology induced by π, the orbit space. We will denote it by X X/G, provided that no confusion occurs. For each point x ∈ X the subset Gx = {g ∈ G | g · x = x} is a closed subgroup of G and is called a stabilizer of x. For each closed subgroup H < G let us consider the following subsets X: XH = {x ∈ X | H · x = x} = {x ∈ X | H ⊂ Gx } (the set of H-fixed points), XH = {x ∈ X | H = Gx },
X(H) = {x ∈ X | H conjugates with Gx }.
In particular when H is the trivial subgroup {e} or H = G, we have: Xfree = Xe = {x ∈ X | Gx = {e}},
Xf ix = XG = {x ∈ X | Gx = G}.
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Let F be a family of orbit types (that is, a subset of the set ConjG of conjugacy classes of closed subgroups of G). Then the set XF {x | (Gx ) ∈ F} ⊂ X is called an F-orbit bundle of X. We say that the G-space X has an orbit type F, or briefly X is the G-F-space, provided that X = XF . Note that metric G-spaces of the orbit type F and G-maps between them generate the category, which is denoted by GF -TOP (or EQUIVF -TOP if all are clear about the group G in question). We will freely use the symbol “G-” or “Equiv-” meaning equivariant. If “∗ ∗ ∗” is any notion from nonequivariant topology, then “G-∗ ∗ ∗” or “Equiv-∗ ∗ ∗” means the corresponding equivariant analogue. The equivariant map f : X → Y is called isovariant if f preserves stabilizers, that is, Gx = Gf (x) for every x ∈ X. The category generated by metric G-spaces of orbit type F and isovariant maps is denoted by ISOVF -TOP (it is always clear about the group G in question). Let C be a category ISOVF -TOP or EQUIVF -TOP. An object of the category C is called an absolute neighbourhood C-extensor (briefly, X ∈ C-ANE) if for each morphism ϕ : A → X of C, defined on a closed G-subset A ⊂ Z of a G-space Z and called a partial C-morphism, can be extended on some G-neighbourhood U ⊂ Z of A to a morphism ϕ : U → X ∈ C. If it is always possible to make U equal to Z, then X is called an absolute C-extensor, X ∈ C-AE. If C coincides with the category EQUIV-TOP (ISOV-TOP), then absolute [neighbourhood] C-extensors are called equivariant [neighbourhood] extensors (isovariant [neighbourhood] extensors) or, briefly, EquivA[N]E-spaces (briefly, Isov-A[N]E-spaces). In what follows we will denote by EquivF -A[N]E (IsovF -A[N]E) the class of injective objects of the category EQUIVF -TOP (ISOVF -TOP). As a example of Isov-AE-space we present W = C(G, L) ∈ Isov-AE where L is a Hilbert space (see [5]). It is easy to see that for each F ⊂ ConjG , the F-orbit bundle WF of W is a universal F-space in the sense of tom Dieck (that is, each F-space X admits, up to G-homotopy, a unique G-map to WF ). It also turns out that the orbit projection p : W → W is partitioned on the stratums {pH : WH → W(H) | H < G} which are the universal N (H)/H-bundles. In particular, the orbit projection pe : Wfree → Wfree is the universal principal G-bundle. 3. The tom Dieck cohomology Let W ∈ Isov-AE. We denote by E(X), where X ∈ Equiv-ANE, the following subspace {(w, x) | Gw < Gx } ⊂ W × X. It turns out that E(X) is an Isov-ANE-space having the G-homotopy type equal to that of X, and the natural projections p : E(X) → W and q : E(X) → X onto the factors are locally Isov-soft and Equiv-soft map, correspondingly (see [2]). The homotopy Borel functor-E : EQUIV-HOMOT → ISOV-HOMOT is defined as follows: X → E(X), and the G-homotopy class of the isovariant map Ef : E(X) → E(Y), (Ef )(w, x) = (w, f (x)), is associated with that of the G-map f : X → Y. (By EQUIV-HOMOT we mean the category, whose objects are Equiv " = ANE-spaces and morphisms are G-homotopy classes of equivariant maps, and we denote by ISOV-HOMOT the category, whose objects are Isov-ANE-spaces and morphisms are isovariant homotopy classes of isovariant maps.) Since the homotopy Borel functor is an equivalence, the passage from X to E(X) occurs without loss of information on a G-homotopy type, but there appears a possibility to draw the additional structure of isovariant extensor for investigation of E(X). In doing so, the map p˜ : E(X) → W of orbit spaces induced by p (which is called a structured map) is locally soft; being considered over W(H) , it represents a bundle with the fibre XH , associated with a universal principal bundle. Since the F-orbit bundle WF of W is a universal F-space in the sense of tom Dieck for each F ⊂ ConjG , (EF × X)/G and EF (X) {(w, x) | Gw < Gx , Gw ∈ F} ⊂ W × X have equal G-homotopy types. Therefore the contravariant functor HF∗ of the (rational) equivariant cohomology with respect to F can be defined as ˇ ˇ ∗ (EF (X); Q) of the Cech the ring HF∗ (X) = H cohomology; and a ring homomorphism HF∗ (f : X → Y) is a ∗ ∗ ∗ ˜f ) : H ˇ (EF (Y ); Q) → H ˇ (EF (X); Q) of the rings H ∗ (Y) and H ∗ (X) which is associated homomorphism (E F F ˜f : EF (X) → EF (Y ) induced by Ef : EF (X) → EF (Y). to the map E
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ˇ ∗ (WF ; Q), it follows that H ∗ (X) Since EF (∗) = {(w, ∗) | Gw < G, (Gw ) ∈ F} = WF and HF∗ (∗) = H F ∗ ∗ is an algebra over the ring HF (∗) = HF (WF ) of the equivariant cohomologies of an F-classifying G-space WF : given r ∈ HF∗ (∗) and x ∈ HF∗ (X), r · x equals the cup product (˜ p)∗ (r) x (see [6]). As it can be easily checked, (EF (G/H))/G = (WF ∩ W≤H )/H and hence the coefficients HF∗ (G/H) of the HF∗ -theory equals ˇ ∗ ((WF ∩ W≤H )/H; Q). H ∗ A partial case of HF∗ (X) is the equivariant Borel cohomology HG (X) [7] corresponding to F = {(e)}; if ∗ F consists of a normal subgroup H < G, then HF (X) coincides with the cohomology of a total space of a bundle with the fibre XH , associated with a universal bundle. The cohomology H ∗ (BG; Q) of the classifying space for G presents one of the many cohomological ∗ ˇ ∗ (Wfree ; Q) = H ∗ (BG; Q), we see that H ∗ (BG; Q) is naturally included invariants of G. Since H{(e)} (∗) = H into a series of cohomological invariants HF∗ (∗) of G parameterized by F ⊂ ConjG . In particular, the case F = P corresponds to the locked cohomology HP∗ (∗). 4. Homomorphisms of cohomologies generated by embeddings of simple semi-ideals In what follows the acting group G will be the two-dimensional torus T2 = S 1 × S 1 identified with the factor group R ⊕ R/Z ⊕ Z, (e2πix , e2πiy ) → (x, y) + Z2 . Note that the set ConjG of conjugacy classes of closed subgroups of T 2 coincides with the set of closed subgroups S = {H < T 2 }, and each proper closed subgroup of T2 is equal to 1. either R · a + Z · b or Z · a + Z · b for appropriate rational vectors a, b ∈ Q × Q (see [8]). We say that the family F ⊂ S is called a semi-ideal if for each H ∈ F and K < H it follows that K ∈ F. The family SH = {K < T 2 | K < H} is called a simple semi-ideal generated by H < T 2 . Let us order the countable set {ai } ⊂ Q × Q of all non-collinear vectors in pairs so that a1 = (1, 0) and a2 = (0, 1). We denote by Hi < T 2 the one-dimensional closed subgroup R · ai and denote by Ii the family {K < T 2 | there is a one-dimensional closed subgroup L < T 2 such that K ∪ Hi ⊂ L}. By the property (1) the family Ii coincides with {R · ai + Z · b | b ∈ Q2 }. Hence it follows that 2. the union ∪{Ii | i ∈ N} coincides with the family P of all proper closed subgroups of T2 . Since the vectors {ai } are non-collinear in pairs, the families Ii are pairwise distinct. Lemma 4.1. If i = j, then Ii ∩ Ij is equal to family D of all discrete subgroups of T2 (see [8]). We say that the families C and C of closed subgroups of T 2 are isomorphic if there is a continuous automorphism χ : T 2 → T 2 such that H ∈ C ⇔ χ(H) ∈ C . We say also that a closed subgroup H < T 2 is standard if H ∈ I1 . It follows by (1) that 3. each family Ii is isomorphic to the standard family I1 , and hence, each proper closed subgroup of T 2 is isomorphic to a standard subgroup. Before deriving our approach on how to calculate the locked cohomologies of the torus we need some facts. Definition 1. A homomorphism L : Zm → Zn , m ≥ n, of Z-modules given by an integer matrix A of maximal rank n is called an almost epimorphism.
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It is clear that each epimorphism is an almost epimorphism, but the opposite is wrong. The following statement supplies a criterion on almost epimorphity. Lemma 4.2. A homomorphism L : Zm → Zn , m ≥ n, is an almost epimorphism if and only if the linear operator ϕ : Hom(Zn , Q) → Hom(Zm , Q) of vector spaces over the field Q adjoint to L is injective. Corollary 1. If a homomorphism L : Zm → Zn , m ≥ n, has finite cokernel, then L is an almost epimorphism. Proof of Lemma 4.2. Since ϕ is an adjoint operator to L, the matrix of ϕ is AT . In view of dimQ (Hom(Zn , Q)) = dimZ (Zn ) = n and rank A = rank(AT ), we can easily deduce that rank A = n if and only if dim(Im(ϕ)) = n, that is ϕ is injective. 2 Lemma 4.3. Let χ : T 2 → T 2 be a continuous automorphism and W ∈ Isov-AE. Then there exists a homotopy equivalence h : W → W such that h(WH ) = Wχ(H) and h : WH → Wχ(H) is a homotopy equivalence for each subgroup H < T 2 . Proof of Lemma 4.3. We consider the new action g ∗ w χ−1 (g)w of T 2 on W, and denote the arising T 2 -space by W . Since W ∈ Isov-AE, there exists an isovariant map f : W → W. It is easy to show that the induced map f˜ : W = W → W of the orbit spaces is the required map h. 2 By the property (1) it follows that each pair L < K < T 2 of proper closed subgroups is isomorphic to the pair of standard subgroups. Then Lemma 4.3 implies the following. Lemma 4.4. If the pair L < K < T 2 of proper closed subgroups is isomorphic to the pair L1 < K1 < T 2 of standard subgroups, then the orbit space WSK is homotopy equivalent to the orbit space WSK1 , and the pair (WSK , WSL ) of orbit spaces is homotopy equivalent to the pair (WSK1 , WSL1 ) of orbit spaces. The homotopy type of the orbit space WSK for the standard subgroup S 1 ⊕ Zn (Zm ⊕ Zn , respectively) was determined in [9]: WSK is the Eilenberg–MacLane space K(Z, 2) (K(Z ⊕ Z, 2), respectively). Thus we have ˇ ∗ (WS , Q), where H = S 1 ⊕ Zn , is isomorphic to the polynomial ring Lemma 4.5. The cohomology ring H H ˇ ∗ (WS , Q), H = Zm ⊕ Zn , is isomorphic to the polynomial ring Q[x] with generator x in degree 2, and H H Q[x, y] with generators x and y in degree 2. Lemma 4.6. The homomorphism L : π2 (WSL ) → π2 (WSK ) of homotopy groups generated by the embedding of the simple semi-ideals SL ⊂ SK , L < K T 2 , is an almost epimorphism. If dim K = dim L, then L is an isomorphism. Proof of Lemma 4.6. By [9] it follows that the orbit space WSK is the base space of the universal bundle π : WSK /K → WSK with the fibre T2 /K, and moreover WSK /K is contractible. In this case the exact homotopy sequence of the bundle gives the isomorphism π2 (WSK ) ∼ = π1 (T2 /K); analogously we have 2 π2 (WSL ) ∼ = π1 (T /L). The natural embedding e : WSL → WSK generates the morphism of the bundles WSL /L
e
π
WSL
WSK /K , π
→
W SK
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which in turn generates a morphism of exact sequences of homotopy groups. As a result, we obtain the following commutative diagram: π2 (WSL )
L
∼ =
π1 (T2 /L)
π2 (WSK ) ∼ =
L
π1 (T2 /K)
we consider the natural bundle T2 /L → T2 /K To determine the properties of the homomorphism L with the fibre K/L and the exact homotopy sequence: 0
π1 (K/L)
π1 (T2 /L)
L
π1 (T2 /K)
π0 (K/L)
0 .
Since π1 (T2 /A) and Z2−dim A are isomorphic for any subgroup A < T2 , and the group π0 (K/L) of : Z2−dim L → components of linear connectivity is finite, it follows by Corollary 1 that the homomorphism L Z2−dim K is an almost epimorphism. is an isomorphism. 2 If dim K = dim L, then π1 (K/L) = 0, and therefore L Theorem 2. Let A be a matrix of the almost epimorphism L : π2 (WSL ) → π2 (WSK ) of homotopy groups generated by the embedding of the simple semi-ideals SL ⊂ SK , L < K T 2 . Then the linear operator ˇ 2 (WS , Q) → H ˇ 2 (WS , Q) of vector spaces is injective, and the matrix of the operator α2 coincides α2 : H K L with the transposed matrix of L. If dim K = dim L, then α2 is an isomorphism. Proof of Theorem 2. The universal coefficient theorem (see [10, p. 128]) gives the following isomorphisms ˇ 2 (WS , Q) ∼ ˇ 2 (WS , Q) ∼ H = Hom(H2 (WSK ), Q) and H = Hom(H2 (WSL ), Q), K L and the Hurewicz theorem (see [10, p. 119]) in view of Lemma 4.5 allows to identify the homology and homotopy groups in the dimension two, that is, H2 (WSL ) ∼ = π2 (WSL ) and H2 (WSK ) ∼ = π2 (WSK ). With 2 ∼ ˇ 2 (WS , Q) ∼ ˇ these identifications, we have H (W ), Q) and H (W , Q) Hom(π Hom(π = = 2 SK SL 2 (WSL ), Q), and K the embedding WSL ⊂ WSK induces the following commutative diagram: ˇ 2 (WS , Q) H K
α2
∼ =
Hom(π2 (WSK ), Q)
ˇ 2 (WS , Q) H L ∼ =
ϕ
Hom(π2 (WSL ), Q),
where the linear operator ϕ adjoint to L. Since by Lemma 4.6 the homomorphism L is an almost epimorphism (L is an isomorphism, provided that dim K = dim L), the operator ϕ is by Lemma 4.2 injective (the operator ϕ is an isomorphism, respectively). 2 Theorem 3. The embedding of the simple semi-ideals SL ⊂ SK , where L < K < T 2 is a pair of proper ˇ ∗ (WS , Q) → H ˇ ∗ (WS , Q) of graded Q-algebras. If dim K = subgroups, induces a monomorphism α∗ : H K L dim L, then α∗ is an isomorphism of graded Q-algebras. Proof of Theorem 3. By Lemma 4.5 the cohomology rings WSL and WSK of the orbit spaces are isomorphic to the polynomial rings on (2 −dim L) and (2 −dim K) generators, respectively. Since the generators of these rings are contained in the second cohomology group, the application of Theorem 2 completes the proof. 2
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The family Ii = {R1 · ai ⊕ Z · b | b ∈ Q2 } contains a unique subgroup of T 2 isomorphic to S 1 ⊕ Zm (for brevity, we denote it by Si1 ⊕ Zm ), and, in its turn, this subgroup contains a unique subgroup Zn ⊕ Zm < Si1 ⊕ Zm . It is clear that the simple semi-ideal SSi1 ⊕Zn can be embedded into the semi-ideal SSi1 ⊕Zm if n divides m. The similar conclusion is valid for the simple semi-ideals SZn1 ⊕Zn2 and SZm1 ⊕Zm2 (here ni divides mi ). From Lemma 4.5 and Theorem 3 it follows easily the following fact: Theorem 4. The embeddings α : WSS1 → WSS1 ⊕Zm and β : W{e} → WSZm1 ⊕Zm2 of orbit spaces induce the i i isomorphisms ˇ ∗ (WS 1 ˇ ∗ (WS 1 , Q) and β ∗ : H ˇ ∗ (WS ˇ ∗ (W{e} , Q) α∗ : H , Q) → H , Q) → H Zm1 ⊕Zm2 S ⊕Zm S i
i
of graded Q-algebras. 5. The cohomology of WIi and WD To proceed the computation of the cohomology of WIi , i ∈ N, and WD we denote by An (Bn , respectively) the simple semi-ideal SSi1 ⊕Zn! (SZn! ⊕Zn! , respectively) and represent the spaces WIi and WD as a growing sequence of open subspaces α2
αn−1
α3
αn
β2
βn−1
β3
WB2 → WB3 → . . . → WBn
{WAn+1 | n ∈ N}, and βn → . . . , WD = {WBn+1 | n ∈ N}.
WA2 → WA3 → . . . → WAn → . . . , WIi =
∗ ˇ ∗ (WA , Q) → H ˇ ∗ (WA , Q) and β ∗ : H ˇ ∗ (WB , Q) → H ˇ ∗ (WB , Q) The linear operators αn−1 :H n−1 n n−1 n n−1 generated by αn−1 and βn−1 are isomorphisms of graded Q-algebras (see Theorem 4), and they generate a tower of vector spaces and linear operators: α∗
α∗ n−1
α∗
α∗
2 3 ˇ q (WA , Q) ←−− ˇ q (WA , Q) ←−− ˇ q (WA , Q) ←−n− . . .} and {H H . . . ←−−−− H 2 3 n ∗
∗
β∗
∗
β2 β3 βn n−1 ˇ q (WB , Q) ← ˇ q (WB , Q) ← ˇ q (WB , Q) ← −− H −− . . . ←−−− H −− . . .}. {H 2 3 n
Theorem 5. For each i ∈ N the following equalities of vector spaces over Q take place: ˇ q (WI , Q) = H i
Q, if q = 2k; 0, if q = 2k + 1,
ˇ q (WD , Q) = and H
Qk+1 , if q = 2k; 0, if q = 2k + 1.
ˇ ∗ (WI , Q) → H ˇ ∗ (WS 1 , Q) and H ˇ ∗ (WD , Q) → H ˇ ∗ (W{e} , Q), induced by Moreover, the linear operators H i Si the embeddings WSS1 → WIi and W{e} → WD , are isomorphisms. i
Proof of Theorem 5. Since the tower of vector spaces and linear operators satisfies the Mittag-Leffler conˇ q−1 (WA , Q)} and lim1 {H ˇ q−1 (WB , Q)} are equal dition (see [11, p. 1229]), the derived functors lim1 {H n n ← ← q q ˇ ˇ to 0. Therefore the vector spaces H (WIi , Q) and H (WD , Q) of cohomologies are the inverse limits ˇ q (WA , Q), αn∗ } and lim{H ˇ q (WB , Q), βn∗ }. Since by Theorem 4 the linear maps αn∗ and βn∗ are isolim{H ←
n
←
n
morphisms, we are able with help of Lemma 4.5 to find for even q the inverse limits ∼ ˇ q (WA , Q), α∗ } ∼ ˇq ˇq lim{H n = H (WA2 , Q) = H (WSS 1 , Q) = Q, and n ←
i
∼ 2 +1 . ˇ q (WB , Q), β ∗ } ∼ ˇq ˇq lim{H n = H (WB2 , Q) = H (W{e} , Q) = Q n q
←
The remaining part of the theorem follows in an obvious way. 2
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ˇ q (WI , Q) → H ˇ q (WD , Q) be a linear operator of cohomology spaces induced by ei : WD → WI , Let ei : H i i q ˇ ˇ and gi : H (WSS1 , Q) → H q (W{e} , Q) a linear operator of cohomology spaces induced by gi : W{e} → WSS1 . i
i
ˇ q (WI , Q) → H ˇ q (WD , Q) is injective for all q ≥ 0. Theorem 6. The linear operator ei : H i Proof of Theorem 6. The following diagram ei
ˇ q (WI , Q) H i
ˇ q (WD , Q) H
ρ
ρ gi
ˇ q (WS 1 , Q) H S
ˇ q (W{e} , Q) H
i
which is commutative (here the vertical morphisms generated by the corresponding embeddings are by Theorem 5 isomorphisms). Theorem 3 implies that gi is an injective operator of cohomology spaces. 2 We consider the subgroups S11 ⊕Zm and Zn ⊕Zm < S11 ⊕Zm from the family I1 = {R1 ·a1 ⊕Z ·b | b ∈ Q2 } and subgroups S21 ⊕Zm and Zn ⊕Zm < S21 ⊕Zm from the family I2 = {R1 ·a2 ⊕Z·b | b ∈ Q2 }, where a1 = (1, 0) ˇ q (WI , Q) → H ˇ q (WD , Q) and a2 = (0, 1). The embeddings D ⊂ Ii induce the injective operators e1 : H 1 q q q q ˇ ˇ ˇ ˇ and e2 : H (WI2 , Q) → H (WD , Q). Let us define its difference e2 − e1 : H (WI1 , Q) ⊕ H (WI2 , Q) → ˇ q (WD , Q) by the formula H ˇ q (WI , Q) and γ2 ∈ H ˇ q (WI , Q). (e2 − e1 )(γ1 , γ2 ) = e2 (γ2 ) − e1 (γ1 ), γ1 ∈ H 1 2 ˇ q (WI , Q) ⊕ H ˇ q (WI , Q) → H ˇ q (WD , Q) is injective for all q ≥ 1. Theorem 7. The linear operator e2 − e1 : H 1 2 Proof of Theorem 7. We consider the following commutative diagram which is the difference of two diagrams from the proof of Theorem 6 (the vertical operators ρ are by Theorem 5 isomorphisms): ˇ q (WI , Q) ⊕ H ˇ q (WI , Q) H 1 2
e2 (γ2 )−e1 (γ1 )
ρ
ρ⊕ρ
ˇ q (WS 1 , Q) ⊕ H ˇ q (WS 1 , Q) H S S 1
ˇ q (WD , Q) H
g2 (t2 )−g1 (t1 )
2
ˇ q (WZ , Q). H 1
ˇ 2 (WS 1 , Q) → H ˇ 2 (WZ , Q), i = 1, 2, is generated by appropriate embedding Here the operator gi : H 1 Si WZ1 → WSS1 . By Theorem 2 the matrix of gi is the matrix transposed to the matrix Ai of the almost i epimorphism Li : π2 (WZ1 ) → π2 (WSS1 ). To calculate Ai we make use of the exact homotopy sequence of i
the bundle T2 → T2 /Si1 from Lemma 4.6: 0
π1 (Si1 )
σi
i L
π1 (T2 )
π1 (T2 /Si1 )
π0 (Si1 )
0,
that is equivalent to 0
Z
σi
Z2
i L
Z
0.
Since the map σ1 : π1 (S11 ) → π1 (T2 ) transforms x into (x,
0), A1 = (0, 1). Analogously, A2 = (1, 0). Then 0 1 the matrix of the linear difference operator g2 −g1 is A = . −1 0
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ˇ 2 (WS 1 , Q) of This result can be interpreted as follows: the operator g1 maps the generator t1 ∈ H S1 ˇ ∗ (WS 1 , Q) ∼ ˇ ∗ (WZ , Q) ∼ ˇ 2 (WZ , Q) of H H ] to the generator y ∈ H y], and the operator g2 maps Q[t Q[x, = = 1 1 1 S1 2 ∗ 2 ∗ ∼ ˇ ˇ ˇ ˇ the generator t2 ∈ H (WS 1 , Q) of H (WS 2 , Q) = Q[t2 ] to the generator x ∈ H (WZ , Q) of H (WZ , Q) ∼ = S2
1
S1
1
ˇ ∗ (WS 1 , Q) → H ˇ ∗ (WZ , Q), i = 1, 2, are homomorphisms Q[x, y] (see Lemma 4.5). Since the operators gi : H 1 S i
of graded Q-algebras, they commute with the cohomology multiplication and thus gi (αtqi ) = α gi (ti )q , ti ∈ Q[ti ]. Therefore the linear operator ˇ 2q (WS 1 , Q) ⊕ H ˇ 2q (WS 1 , Q) → H ˇ 2q (WZ , Q) g2 −g1 : H 1 S S 1
2
of cohomology spaces in high dimensions acts as follows: {αtq1 } ⊕{βtq2 } ∈ Q[t1 ] ⊕Q[t2 ] → {α(−1)q y q +βxq } ∈ Q[x, y], and hence it is injective. 2 6. The proof of Theorem 1 By the property (2) the family P of all proper closed subgroups of T2 coincides with ∪{Ii | i ∈ N}. For ˇ q (WI , Q), Iq H ˇ q (W N , Q), N > 1, and simplicity sake, we introduce the following notations: Iq1,i H i N ∪ Ii
i=1
ˇ q (WD , Q). Then the locked cohomology H ˇ q (WP , Q) of the torus coincides with the inverse limit D H q q lim{IN , fN }, where the operator q
←
q ˇ q (W N fN : IqN = H
∪ Ii
ˇ q (WN −1 , Q) , Q) → IqN−1 = H ∪ Ii
i=1
is induced by the inclusion WN −1
∪ Ii
⊂WN
∪ Ii
i=1
.
i=1
i=1
⎧ ⎪ ⎪ ⎨Q, if q = 0 q Lemma 6.1. I2 = 0, if q = 1 or q = 2k, k ∈ N ⎪ ⎪ ⎩Q q−1 2 −1 , if q = 2k + 1, k ∈ N. Proof of Lemma 6.1. Since by Lemma 4.1 WI1 ∩ WI2 = WD , the Mayer–Vietoris sequence for WI1 and WI2 (see [6]) is as follows: e2k −e2k
δ
δ
2 1 2k 2k ... − → D2k−1 − → I2k → I2k − → I2k+1 → I2k+1 − ⊕ I2k+1 → ..., − 2 − 1,1 ⊕ I1,2 −−−−−→ D 2 1,1 1,2
2k where e2k 2 −e1 is the difference of the operators induced by the corresponding embeddings. Theorem 7 implies that this difference is an injective operator for all k > 0. In view of dim D0 = 1 and Theorem 6 this difference for k = 0 is an epimorphism. 2k+1 = 0. Hence the With help of Theorem 5 we know that D2k = Qk+1 , D2k+1 = 0, I2k 1,i = Q and I1,i Mayer–Vietoris sequence splits into a series of short exact sequences: if k = 0, then epi
δ
δ
... − →0− → I02 − → Q ⊕ Q −−→ Q − → I12 − →0− → ..., and if k > 0, then δ
mono
δ
... − →0− → I2k → Q ⊕ Q −−−−→ Qk+1 − → I2k+1 →0− − → ... 2 − 2 As a result, we obtain the desired cohomology Iq2 .
2
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⎧ ⎪ ⎪ ⎨Q, if q = 0 q Theorem 8. Given N > 1, IN = 0, if q = 1 or q = 2k, k ∈ N ⎪ ⎪ ⎩Q(N −1) q−1 2 −1 , if q = 2k + 1, k ∈ N. Proof of Theorem 8. will be performed by induction on N . The basis is valid by Lemma 6.1. We assume that for all numbers less than N, N > 2, the theorem holds, and show that for N it remains valid. Since by Lemma 4.1 WN −1 ∩ WIN = D, the Mayer–Vietoris sequence for W N = WN −1 ∪ WIN ∪ Ii
∪ Ii
i=1
i=1
∪ Ii
i=1
looks like as follows: e2k −j
δ
f 2k+1 ⊕0
δ
2k+1 2k ... − → D2k−1 − → I2k → I2k −2−−→ D2k − → I2k+1 −−N−−−−→ I2k+1 → D2k+1 − → .... N − N−1 ⊕ I1,N − N N−1 ⊕ I1,N − 2k 2k Here the operator e2k is a linear operator induced by WD → 2 is injective by Theorem 6, j : IN−1 → D 2k+1 2k+1 2k+1 is a linear operator from IN to IN−1 induced by WN −1 → W N . Since D2k+1 = 0, WN −1 , and fN ∪ Ii
∪ Ii
i=1
i=1
∪ Ii
i=1
this sequence can be written as follows: epi
δ
δ
... − →0− → I0N − → Q ⊕ Q −−→ Q − → I1N − →0− → ..., for k = 0, and f 2k+1
... − →0− → I2k → 0 ⊕ Q −−−−→ Qk+1 − → I2k+1 −−N−−→ Q(N −2)k−1 ⊕ 0 − →0− → ... N − N δ
δ
mono
2k+1 for k > 0. Its analysis implies that I2k = Q(N −1)k−1 for k > 0, and I0N = Q and I1N = 0 N = 0 and IN q−1 q−1 for k = 0. Since q = 2k + 1 and k = 2 , we obtain that IqN = Q(N −1) 2 −1 for odd q > 0. Theorem 8 is proved. 2 2k+1 Corollary 2. Each operator fN : I2k+1 → I2k+1 N N−1 is epimorphism. q ˇ q (WP , Q) and all components of this inverse spectrum are known (in particular, }=H Since lim{IqN , fN ←
q each fN is by Corollary 2 an epimorphism for q = 2k + 1, k > 0), the inverse limit of spectrum fq
fq
fq
fq
q fN +1
N lim{Iq2 ←−3− Iq3 ←−4− Iq4 ←−5− . . . ←− − IqN ←−−− . . .} =
←
lim{Q ←
q−1 2 −1
epi
←−− Q2
q−1 2 −1
epi
←−− Q3
q−1 2 −1
epi
epi
←−− . . . ←−− Q(N −1)
q−1 2 −1
epi
←−− . . .}
is a countable power Qω of Q. Theorem 1 is completely proved. References [1] T. tom Dieck, Transformation Groups and Representation Theory, Lect. Notes Math., vol. 766, Springer-Verlag, Berlin– New York, 1979. [2] S.M. Ageev, On Palais universal G-spaces and isovariant absolute extensors, Mat. Sb. 203 (6) (2012) 3–34. [3] S. Ageev, S. Bogatyi, The Banach–Mazur compactum in the dimension two, Topology Atlas, 1997, Preprint No 291. [4] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York–London, 1972. [5] S.M. Ageev, Exp, Mat. Sb. 203 (6) (2016) 3–34. [6] W.S. Massey, Homology and Cohomology Theory, Monogr. Textb. Pure Appl. Math., vol. 46, Marcel Dekker, Inc., New York, 1978. [7] Wu Yi Hsiang, Cohomology Theory of Topological Transformation Groups, Springer-Verlag, New York, Heidelberg, Berlin, 1975.
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[8] S.A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Lond. Math. Soc. Lect. Note Ser., 1977. [9] S.M. Ageev, I.V. Usimov, The cohomology ring of subspace of universal S 1 -space with finite orbit types, Topol. Appl. 160 (11) (2013) 1255–1260. [10] A.T. Fomenko, D.B. Fuchs, Homotopical Topology, Graduate Texts in Mathematics, 2016, http://www.springer.com/ series/136. [11] I.M. James, C.A. McGibbon, Phantom Maps//Handbook of Algebraic Topology, Elsever, 1995.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Virtual Special Issue – Dedicated to the 120th anniversary of the eminent Russian mathematician P.S. Alexandroff
The locked cohomology of the torus ✩ Igor Usimov, S.M. Ageev ∗ Belarus State University, Minsk, Belarus
a r t i c l e
i n f o
Article history: Received 13 July 2016 Accepted 15 November 2016 Available online 11 February 2017
a b s t r a c t We calculate the ring of the locked cohomology of the torus. © 2017 Elsevier B.V. All rights reserved.
MSC: 55N91 55P20 55M15 55R40 Keywords: Isovariant absolute extensors Universal G-space in the sense of Palais Equivariant tom Dieck cohomologies Universal G-space in the sense of tom Dieck
1. Introduction For a semi-ideal F ⊂ ConjG of orbit types, a universal F-space EF (in the sense of tom Dieck [1]) generates an equivariant cohomology theories HF∗ (called the tom Dieck cohomology theories) which assigns the ring HF∗ (X) = H ∗ ((EF ×X)/G; Q) of the Čech cohomology to X (here Q is the field of rational numbers). A partial case of HF∗ is the equivariant Borel cohomology corresponding F = {(e)}. The tom Dieck cohomology possesses many convenient formal properties, but its computation is difficult (in comparison with the classical Borel equivariant cohomology). However, within the framework of the theory of the universal Palais G-spaces (in other words, the theory of isovariant extensors, or Isov-AE-spaces, see [2]), there appears a new possibility for construction and computation of the tom Dieck cohomology. It turns out that universal F-spaces exist for all families F of orbit types, and all of them are concentrated within an Isov-AE-space W (the so-called effect of concentration): WF is a F-universal space in the sense ✩
This paper was written with the partial support of a grant from the Ministry of Education of the Republic of Belarus.
* Corresponding author. E-mail address: [email protected] (S.M. Ageev). http://dx.doi.org/10.1016/j.topol.2017.02.033 0166-8641/© 2017 Elsevier B.V. All rights reserved.
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of tom Dieck. Moreover, they have the properties of mutual location in W as good as it can be possible. This method allows to bring the computations of some cohomology invariants to the explicit formulae. ˇ ∗ (WF = WF /G; Q), we have a strong need for its Since HF∗ (X) is an algebra over the ring HF∗ (∗) = H calculation. In general, this problem seems to be quite difficult. Even for a compact connected Lie group G, the problem of computation of the cohomology HP∗ (∗) where P = ConjG \{(e)} (which we prefer to call a locked cohomology) is still far from being completely solved, though it presents an considerable interest in the homogeneity conjecture for the Banach–Mazur compactum (in particular, the establishment of its non-triviality) (see [3]). It is known that HP∗ (∗) is the polynomial ring Q[x] with generator x of degree 2 (for G = S 1 ) and 4 (for G = O(2)) (see [2]). The aim of the paper is the computation of the locked cohomology for G = T2 = S 1 × S 1 . ⎧ ⎪ ⎪ ⎨Q, if q = 0 q ˇ (WP , Q) = 0, if q = 1 or q = 2k, k ∈ N Theorem 1. H ⎪ ⎪ ⎩Qω , if q = 2k + 1, k ∈ N,
where Qω denotes the countable power of Q.
Theorem 1 implies that the multiplication in HP∗ (∗) is trivial. 2. Preliminary facts and results In what follows we shall assume all spaces (all maps) to be metric (continuous, respectively), if they do not arise as a result of some constructions or if the opposite is not claimed; all acting groups are assumed to be compact Lie groups. We present the basic notions of the theory of G-spaces [4]. An action of a compact group G on a space X is a continuous map μ from the product G × X into X satisfying the following properties: 1. μ(g, μ(h, x)) = μ(g · h, x); and 2. μ(e, x) = x for all x ∈ X, g, h ∈ G (here e is the unit of the group G). As a rule, μ(g, x) will be written as g · x or just gx. A space X with an action of the group G is called a G-space. The map f : X → Y of G-spaces is called a G-map or an equivariant map if f (g · x) = g · f (x) for all x ∈ X, g ∈ G. The subset {g · x | g ∈ G} = G · x is called the orbit G(x) of the point x ∈ X which turns out to be closed. The natural map π = πX : X → X/G, x → G(x), of the space X onto the space X/G of quotient partition is said to be the orbit projection. We call the space of quotient partition, equipped with the quotient topology induced by π, the orbit space. We will denote it by X X/G, provided that no confusion occurs. For each point x ∈ X the subset Gx = {g ∈ G | g · x = x} is a closed subgroup of G and is called a stabilizer of x. For each closed subgroup H < G let us consider the following subsets X: XH = {x ∈ X | H · x = x} = {x ∈ X | H ⊂ Gx } (the set of H-fixed points), XH = {x ∈ X | H = Gx },
X(H) = {x ∈ X | H conjugates with Gx }.
In particular when H is the trivial subgroup {e} or H = G, we have: Xfree = Xe = {x ∈ X | Gx = {e}},
Xf ix = XG = {x ∈ X | Gx = G}.
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Let F be a family of orbit types (that is, a subset of the set ConjG of conjugacy classes of closed subgroups of G). Then the set XF {x | (Gx ) ∈ F} ⊂ X is called an F-orbit bundle of X. We say that the G-space X has an orbit type F, or briefly X is the G-F-space, provided that X = XF . Note that metric G-spaces of the orbit type F and G-maps between them generate the category, which is denoted by GF -TOP (or EQUIVF -TOP if all are clear about the group G in question). We will freely use the symbol “G-” or “Equiv-” meaning equivariant. If “∗ ∗ ∗” is any notion from nonequivariant topology, then “G-∗ ∗ ∗” or “Equiv-∗ ∗ ∗” means the corresponding equivariant analogue. The equivariant map f : X → Y is called isovariant if f preserves stabilizers, that is, Gx = Gf (x) for every x ∈ X. The category generated by metric G-spaces of orbit type F and isovariant maps is denoted by ISOVF -TOP (it is always clear about the group G in question). Let C be a category ISOVF -TOP or EQUIVF -TOP. An object of the category C is called an absolute neighbourhood C-extensor (briefly, X ∈ C-ANE) if for each morphism ϕ : A → X of C, defined on a closed G-subset A ⊂ Z of a G-space Z and called a partial C-morphism, can be extended on some G-neighbourhood U ⊂ Z of A to a morphism ϕ : U → X ∈ C. If it is always possible to make U equal to Z, then X is called an absolute C-extensor, X ∈ C-AE. If C coincides with the category EQUIV-TOP (ISOV-TOP), then absolute [neighbourhood] C-extensors are called equivariant [neighbourhood] extensors (isovariant [neighbourhood] extensors) or, briefly, EquivA[N]E-spaces (briefly, Isov-A[N]E-spaces). In what follows we will denote by EquivF -A[N]E (IsovF -A[N]E) the class of injective objects of the category EQUIVF -TOP (ISOVF -TOP). As a example of Isov-AE-space we present W = C(G, L) ∈ Isov-AE where L is a Hilbert space (see [5]). It is easy to see that for each F ⊂ ConjG , the F-orbit bundle WF of W is a universal F-space in the sense of tom Dieck (that is, each F-space X admits, up to G-homotopy, a unique G-map to WF ). It also turns out that the orbit projection p : W → W is partitioned on the stratums {pH : WH → W(H) | H < G} which are the universal N (H)/H-bundles. In particular, the orbit projection pe : Wfree → Wfree is the universal principal G-bundle. 3. The tom Dieck cohomology Let W ∈ Isov-AE. We denote by E(X), where X ∈ Equiv-ANE, the following subspace {(w, x) | Gw < Gx } ⊂ W × X. It turns out that E(X) is an Isov-ANE-space having the G-homotopy type equal to that of X, and the natural projections p : E(X) → W and q : E(X) → X onto the factors are locally Isov-soft and Equiv-soft map, correspondingly (see [2]). The homotopy Borel functor-E : EQUIV-HOMOT → ISOV-HOMOT is defined as follows: X → E(X), and the G-homotopy class of the isovariant map Ef : E(X) → E(Y), (Ef )(w, x) = (w, f (x)), is associated with that of the G-map f : X → Y. (By EQUIV-HOMOT we mean the category, whose objects are Equiv " = ANE-spaces and morphisms are G-homotopy classes of equivariant maps, and we denote by ISOV-HOMOT the category, whose objects are Isov-ANE-spaces and morphisms are isovariant homotopy classes of isovariant maps.) Since the homotopy Borel functor is an equivalence, the passage from X to E(X) occurs without loss of information on a G-homotopy type, but there appears a possibility to draw the additional structure of isovariant extensor for investigation of E(X). In doing so, the map p˜ : E(X) → W of orbit spaces induced by p (which is called a structured map) is locally soft; being considered over W(H) , it represents a bundle with the fibre XH , associated with a universal principal bundle. Since the F-orbit bundle WF of W is a universal F-space in the sense of tom Dieck for each F ⊂ ConjG , (EF × X)/G and EF (X) {(w, x) | Gw < Gx , Gw ∈ F} ⊂ W × X have equal G-homotopy types. Therefore the contravariant functor HF∗ of the (rational) equivariant cohomology with respect to F can be defined as ˇ ˇ ∗ (EF (X); Q) of the Cech the ring HF∗ (X) = H cohomology; and a ring homomorphism HF∗ (f : X → Y) is a ∗ ∗ ∗ ˜f ) : H ˇ (EF (Y ); Q) → H ˇ (EF (X); Q) of the rings H ∗ (Y) and H ∗ (X) which is associated homomorphism (E F F ˜f : EF (X) → EF (Y ) induced by Ef : EF (X) → EF (Y). to the map E
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ˇ ∗ (WF ; Q), it follows that H ∗ (X) Since EF (∗) = {(w, ∗) | Gw < G, (Gw ) ∈ F} = WF and HF∗ (∗) = H F ∗ ∗ is an algebra over the ring HF (∗) = HF (WF ) of the equivariant cohomologies of an F-classifying G-space WF : given r ∈ HF∗ (∗) and x ∈ HF∗ (X), r · x equals the cup product (˜ p)∗ (r) x (see [6]). As it can be easily checked, (EF (G/H))/G = (WF ∩ W≤H )/H and hence the coefficients HF∗ (G/H) of the HF∗ -theory equals ˇ ∗ ((WF ∩ W≤H )/H; Q). H ∗ A partial case of HF∗ (X) is the equivariant Borel cohomology HG (X) [7] corresponding to F = {(e)}; if ∗ F consists of a normal subgroup H < G, then HF (X) coincides with the cohomology of a total space of a bundle with the fibre XH , associated with a universal bundle. The cohomology H ∗ (BG; Q) of the classifying space for G presents one of the many cohomological ∗ ˇ ∗ (Wfree ; Q) = H ∗ (BG; Q), we see that H ∗ (BG; Q) is naturally included invariants of G. Since H{(e)} (∗) = H into a series of cohomological invariants HF∗ (∗) of G parameterized by F ⊂ ConjG . In particular, the case F = P corresponds to the locked cohomology HP∗ (∗). 4. Homomorphisms of cohomologies generated by embeddings of simple semi-ideals In what follows the acting group G will be the two-dimensional torus T2 = S 1 × S 1 identified with the factor group R ⊕ R/Z ⊕ Z, (e2πix , e2πiy ) → (x, y) + Z2 . Note that the set ConjG of conjugacy classes of closed subgroups of T 2 coincides with the set of closed subgroups S = {H < T 2 }, and each proper closed subgroup of T2 is equal to 1. either R · a + Z · b or Z · a + Z · b for appropriate rational vectors a, b ∈ Q × Q (see [8]). We say that the family F ⊂ S is called a semi-ideal if for each H ∈ F and K < H it follows that K ∈ F. The family SH = {K < T 2 | K < H} is called a simple semi-ideal generated by H < T 2 . Let us order the countable set {ai } ⊂ Q × Q of all non-collinear vectors in pairs so that a1 = (1, 0) and a2 = (0, 1). We denote by Hi < T 2 the one-dimensional closed subgroup R · ai and denote by Ii the family {K < T 2 | there is a one-dimensional closed subgroup L < T 2 such that K ∪ Hi ⊂ L}. By the property (1) the family Ii coincides with {R · ai + Z · b | b ∈ Q2 }. Hence it follows that 2. the union ∪{Ii | i ∈ N} coincides with the family P of all proper closed subgroups of T2 . Since the vectors {ai } are non-collinear in pairs, the families Ii are pairwise distinct. Lemma 4.1. If i = j, then Ii ∩ Ij is equal to family D of all discrete subgroups of T2 (see [8]). We say that the families C and C of closed subgroups of T 2 are isomorphic if there is a continuous automorphism χ : T 2 → T 2 such that H ∈ C ⇔ χ(H) ∈ C . We say also that a closed subgroup H < T 2 is standard if H ∈ I1 . It follows by (1) that 3. each family Ii is isomorphic to the standard family I1 , and hence, each proper closed subgroup of T 2 is isomorphic to a standard subgroup. Before deriving our approach on how to calculate the locked cohomologies of the torus we need some facts. Definition 1. A homomorphism L : Zm → Zn , m ≥ n, of Z-modules given by an integer matrix A of maximal rank n is called an almost epimorphism.
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It is clear that each epimorphism is an almost epimorphism, but the opposite is wrong. The following statement supplies a criterion on almost epimorphity. Lemma 4.2. A homomorphism L : Zm → Zn , m ≥ n, is an almost epimorphism if and only if the linear operator ϕ : Hom(Zn , Q) → Hom(Zm , Q) of vector spaces over the field Q adjoint to L is injective. Corollary 1. If a homomorphism L : Zm → Zn , m ≥ n, has finite cokernel, then L is an almost epimorphism. Proof of Lemma 4.2. Since ϕ is an adjoint operator to L, the matrix of ϕ is AT . In view of dimQ (Hom(Zn , Q)) = dimZ (Zn ) = n and rank A = rank(AT ), we can easily deduce that rank A = n if and only if dim(Im(ϕ)) = n, that is ϕ is injective. 2 Lemma 4.3. Let χ : T 2 → T 2 be a continuous automorphism and W ∈ Isov-AE. Then there exists a homotopy equivalence h : W → W such that h(WH ) = Wχ(H) and h : WH → Wχ(H) is a homotopy equivalence for each subgroup H < T 2 . Proof of Lemma 4.3. We consider the new action g ∗ w χ−1 (g)w of T 2 on W, and denote the arising T 2 -space by W . Since W ∈ Isov-AE, there exists an isovariant map f : W → W. It is easy to show that the induced map f˜ : W = W → W of the orbit spaces is the required map h. 2 By the property (1) it follows that each pair L < K < T 2 of proper closed subgroups is isomorphic to the pair of standard subgroups. Then Lemma 4.3 implies the following. Lemma 4.4. If the pair L < K < T 2 of proper closed subgroups is isomorphic to the pair L1 < K1 < T 2 of standard subgroups, then the orbit space WSK is homotopy equivalent to the orbit space WSK1 , and the pair (WSK , WSL ) of orbit spaces is homotopy equivalent to the pair (WSK1 , WSL1 ) of orbit spaces. The homotopy type of the orbit space WSK for the standard subgroup S 1 ⊕ Zn (Zm ⊕ Zn , respectively) was determined in [9]: WSK is the Eilenberg–MacLane space K(Z, 2) (K(Z ⊕ Z, 2), respectively). Thus we have ˇ ∗ (WS , Q), where H = S 1 ⊕ Zn , is isomorphic to the polynomial ring Lemma 4.5. The cohomology ring H H ˇ ∗ (WS , Q), H = Zm ⊕ Zn , is isomorphic to the polynomial ring Q[x] with generator x in degree 2, and H H Q[x, y] with generators x and y in degree 2. Lemma 4.6. The homomorphism L : π2 (WSL ) → π2 (WSK ) of homotopy groups generated by the embedding of the simple semi-ideals SL ⊂ SK , L < K T 2 , is an almost epimorphism. If dim K = dim L, then L is an isomorphism. Proof of Lemma 4.6. By [9] it follows that the orbit space WSK is the base space of the universal bundle π : WSK /K → WSK with the fibre T2 /K, and moreover WSK /K is contractible. In this case the exact homotopy sequence of the bundle gives the isomorphism π2 (WSK ) ∼ = π1 (T2 /K); analogously we have 2 π2 (WSL ) ∼ = π1 (T /L). The natural embedding e : WSL → WSK generates the morphism of the bundles WSL /L
e
π
WSL
WSK /K , π
→
W SK
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which in turn generates a morphism of exact sequences of homotopy groups. As a result, we obtain the following commutative diagram: π2 (WSL )
L
∼ =
π1 (T2 /L)
π2 (WSK ) ∼ =
L
π1 (T2 /K)
we consider the natural bundle T2 /L → T2 /K To determine the properties of the homomorphism L with the fibre K/L and the exact homotopy sequence: 0
π1 (K/L)
π1 (T2 /L)
L
π1 (T2 /K)
π0 (K/L)
0 .
Since π1 (T2 /A) and Z2−dim A are isomorphic for any subgroup A < T2 , and the group π0 (K/L) of : Z2−dim L → components of linear connectivity is finite, it follows by Corollary 1 that the homomorphism L Z2−dim K is an almost epimorphism. is an isomorphism. 2 If dim K = dim L, then π1 (K/L) = 0, and therefore L Theorem 2. Let A be a matrix of the almost epimorphism L : π2 (WSL ) → π2 (WSK ) of homotopy groups generated by the embedding of the simple semi-ideals SL ⊂ SK , L < K T 2 . Then the linear operator ˇ 2 (WS , Q) → H ˇ 2 (WS , Q) of vector spaces is injective, and the matrix of the operator α2 coincides α2 : H K L with the transposed matrix of L. If dim K = dim L, then α2 is an isomorphism. Proof of Theorem 2. The universal coefficient theorem (see [10, p. 128]) gives the following isomorphisms ˇ 2 (WS , Q) ∼ ˇ 2 (WS , Q) ∼ H = Hom(H2 (WSK ), Q) and H = Hom(H2 (WSL ), Q), K L and the Hurewicz theorem (see [10, p. 119]) in view of Lemma 4.5 allows to identify the homology and homotopy groups in the dimension two, that is, H2 (WSL ) ∼ = π2 (WSL ) and H2 (WSK ) ∼ = π2 (WSK ). With 2 ∼ ˇ 2 (WS , Q) ∼ ˇ these identifications, we have H (W ), Q) and H (W , Q) Hom(π Hom(π = = 2 SK SL 2 (WSL ), Q), and K the embedding WSL ⊂ WSK induces the following commutative diagram: ˇ 2 (WS , Q) H K
α2
∼ =
Hom(π2 (WSK ), Q)
ˇ 2 (WS , Q) H L ∼ =
ϕ
Hom(π2 (WSL ), Q),
where the linear operator ϕ adjoint to L. Since by Lemma 4.6 the homomorphism L is an almost epimorphism (L is an isomorphism, provided that dim K = dim L), the operator ϕ is by Lemma 4.2 injective (the operator ϕ is an isomorphism, respectively). 2 Theorem 3. The embedding of the simple semi-ideals SL ⊂ SK , where L < K < T 2 is a pair of proper ˇ ∗ (WS , Q) → H ˇ ∗ (WS , Q) of graded Q-algebras. If dim K = subgroups, induces a monomorphism α∗ : H K L dim L, then α∗ is an isomorphism of graded Q-algebras. Proof of Theorem 3. By Lemma 4.5 the cohomology rings WSL and WSK of the orbit spaces are isomorphic to the polynomial rings on (2 −dim L) and (2 −dim K) generators, respectively. Since the generators of these rings are contained in the second cohomology group, the application of Theorem 2 completes the proof. 2
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The family Ii = {R1 · ai ⊕ Z · b | b ∈ Q2 } contains a unique subgroup of T 2 isomorphic to S 1 ⊕ Zm (for brevity, we denote it by Si1 ⊕ Zm ), and, in its turn, this subgroup contains a unique subgroup Zn ⊕ Zm < Si1 ⊕ Zm . It is clear that the simple semi-ideal SSi1 ⊕Zn can be embedded into the semi-ideal SSi1 ⊕Zm if n divides m. The similar conclusion is valid for the simple semi-ideals SZn1 ⊕Zn2 and SZm1 ⊕Zm2 (here ni divides mi ). From Lemma 4.5 and Theorem 3 it follows easily the following fact: Theorem 4. The embeddings α : WSS1 → WSS1 ⊕Zm and β : W{e} → WSZm1 ⊕Zm2 of orbit spaces induce the i i isomorphisms ˇ ∗ (WS 1 ˇ ∗ (WS 1 , Q) and β ∗ : H ˇ ∗ (WS ˇ ∗ (W{e} , Q) α∗ : H , Q) → H , Q) → H Zm1 ⊕Zm2 S ⊕Zm S i
i
of graded Q-algebras. 5. The cohomology of WIi and WD To proceed the computation of the cohomology of WIi , i ∈ N, and WD we denote by An (Bn , respectively) the simple semi-ideal SSi1 ⊕Zn! (SZn! ⊕Zn! , respectively) and represent the spaces WIi and WD as a growing sequence of open subspaces α2
αn−1
α3
αn
β2
βn−1
β3
WB2 → WB3 → . . . → WBn
{WAn+1 | n ∈ N}, and βn → . . . , WD = {WBn+1 | n ∈ N}.
WA2 → WA3 → . . . → WAn → . . . , WIi =
∗ ˇ ∗ (WA , Q) → H ˇ ∗ (WA , Q) and β ∗ : H ˇ ∗ (WB , Q) → H ˇ ∗ (WB , Q) The linear operators αn−1 :H n−1 n n−1 n n−1 generated by αn−1 and βn−1 are isomorphisms of graded Q-algebras (see Theorem 4), and they generate a tower of vector spaces and linear operators: α∗
α∗ n−1
α∗
α∗
2 3 ˇ q (WA , Q) ←−− ˇ q (WA , Q) ←−− ˇ q (WA , Q) ←−n− . . .} and {H H . . . ←−−−− H 2 3 n ∗
∗
β∗
∗
β2 β3 βn n−1 ˇ q (WB , Q) ← ˇ q (WB , Q) ← ˇ q (WB , Q) ← −− H −− . . . ←−−− H −− . . .}. {H 2 3 n
Theorem 5. For each i ∈ N the following equalities of vector spaces over Q take place: ˇ q (WI , Q) = H i
Q, if q = 2k; 0, if q = 2k + 1,
ˇ q (WD , Q) = and H
Qk+1 , if q = 2k; 0, if q = 2k + 1.
ˇ ∗ (WI , Q) → H ˇ ∗ (WS 1 , Q) and H ˇ ∗ (WD , Q) → H ˇ ∗ (W{e} , Q), induced by Moreover, the linear operators H i Si the embeddings WSS1 → WIi and W{e} → WD , are isomorphisms. i
Proof of Theorem 5. Since the tower of vector spaces and linear operators satisfies the Mittag-Leffler conˇ q−1 (WA , Q)} and lim1 {H ˇ q−1 (WB , Q)} are equal dition (see [11, p. 1229]), the derived functors lim1 {H n n ← ← q q ˇ ˇ to 0. Therefore the vector spaces H (WIi , Q) and H (WD , Q) of cohomologies are the inverse limits ˇ q (WA , Q), αn∗ } and lim{H ˇ q (WB , Q), βn∗ }. Since by Theorem 4 the linear maps αn∗ and βn∗ are isolim{H ←
n
←
n
morphisms, we are able with help of Lemma 4.5 to find for even q the inverse limits ∼ ˇ q (WA , Q), α∗ } ∼ ˇq ˇq lim{H n = H (WA2 , Q) = H (WSS 1 , Q) = Q, and n ←
i
∼ 2 +1 . ˇ q (WB , Q), β ∗ } ∼ ˇq ˇq lim{H n = H (WB2 , Q) = H (W{e} , Q) = Q n q
←
The remaining part of the theorem follows in an obvious way. 2
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ˇ q (WI , Q) → H ˇ q (WD , Q) be a linear operator of cohomology spaces induced by ei : WD → WI , Let ei : H i i q ˇ ˇ and gi : H (WSS1 , Q) → H q (W{e} , Q) a linear operator of cohomology spaces induced by gi : W{e} → WSS1 . i
i
ˇ q (WI , Q) → H ˇ q (WD , Q) is injective for all q ≥ 0. Theorem 6. The linear operator ei : H i Proof of Theorem 6. The following diagram ei
ˇ q (WI , Q) H i
ˇ q (WD , Q) H
ρ
ρ gi
ˇ q (WS 1 , Q) H S
ˇ q (W{e} , Q) H
i
which is commutative (here the vertical morphisms generated by the corresponding embeddings are by Theorem 5 isomorphisms). Theorem 3 implies that gi is an injective operator of cohomology spaces. 2 We consider the subgroups S11 ⊕Zm and Zn ⊕Zm < S11 ⊕Zm from the family I1 = {R1 ·a1 ⊕Z ·b | b ∈ Q2 } and subgroups S21 ⊕Zm and Zn ⊕Zm < S21 ⊕Zm from the family I2 = {R1 ·a2 ⊕Z·b | b ∈ Q2 }, where a1 = (1, 0) ˇ q (WI , Q) → H ˇ q (WD , Q) and a2 = (0, 1). The embeddings D ⊂ Ii induce the injective operators e1 : H 1 q q q q ˇ ˇ ˇ ˇ and e2 : H (WI2 , Q) → H (WD , Q). Let us define its difference e2 − e1 : H (WI1 , Q) ⊕ H (WI2 , Q) → ˇ q (WD , Q) by the formula H ˇ q (WI , Q) and γ2 ∈ H ˇ q (WI , Q). (e2 − e1 )(γ1 , γ2 ) = e2 (γ2 ) − e1 (γ1 ), γ1 ∈ H 1 2 ˇ q (WI , Q) ⊕ H ˇ q (WI , Q) → H ˇ q (WD , Q) is injective for all q ≥ 1. Theorem 7. The linear operator e2 − e1 : H 1 2 Proof of Theorem 7. We consider the following commutative diagram which is the difference of two diagrams from the proof of Theorem 6 (the vertical operators ρ are by Theorem 5 isomorphisms): ˇ q (WI , Q) ⊕ H ˇ q (WI , Q) H 1 2
e2 (γ2 )−e1 (γ1 )
ρ
ρ⊕ρ
ˇ q (WS 1 , Q) ⊕ H ˇ q (WS 1 , Q) H S S 1
ˇ q (WD , Q) H
g2 (t2 )−g1 (t1 )
2
ˇ q (WZ , Q). H 1
ˇ 2 (WS 1 , Q) → H ˇ 2 (WZ , Q), i = 1, 2, is generated by appropriate embedding Here the operator gi : H 1 Si WZ1 → WSS1 . By Theorem 2 the matrix of gi is the matrix transposed to the matrix Ai of the almost i epimorphism Li : π2 (WZ1 ) → π2 (WSS1 ). To calculate Ai we make use of the exact homotopy sequence of i
the bundle T2 → T2 /Si1 from Lemma 4.6: 0
π1 (Si1 )
σi
i L
π1 (T2 )
π1 (T2 /Si1 )
π0 (Si1 )
0,
that is equivalent to 0
Z
σi
Z2
i L
Z
0.
Since the map σ1 : π1 (S11 ) → π1 (T2 ) transforms x into (x,
0), A1 = (0, 1). Analogously, A2 = (1, 0). Then 0 1 the matrix of the linear difference operator g2 −g1 is A = . −1 0
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ˇ 2 (WS 1 , Q) of This result can be interpreted as follows: the operator g1 maps the generator t1 ∈ H S1 ˇ ∗ (WS 1 , Q) ∼ ˇ ∗ (WZ , Q) ∼ ˇ 2 (WZ , Q) of H H ] to the generator y ∈ H y], and the operator g2 maps Q[t Q[x, = = 1 1 1 S1 2 ∗ 2 ∗ ∼ ˇ ˇ ˇ ˇ the generator t2 ∈ H (WS 1 , Q) of H (WS 2 , Q) = Q[t2 ] to the generator x ∈ H (WZ , Q) of H (WZ , Q) ∼ = S2
1
S1
1
ˇ ∗ (WS 1 , Q) → H ˇ ∗ (WZ , Q), i = 1, 2, are homomorphisms Q[x, y] (see Lemma 4.5). Since the operators gi : H 1 S i
of graded Q-algebras, they commute with the cohomology multiplication and thus gi (αtqi ) = α gi (ti )q , ti ∈ Q[ti ]. Therefore the linear operator ˇ 2q (WS 1 , Q) ⊕ H ˇ 2q (WS 1 , Q) → H ˇ 2q (WZ , Q) g2 −g1 : H 1 S S 1
2
of cohomology spaces in high dimensions acts as follows: {αtq1 } ⊕{βtq2 } ∈ Q[t1 ] ⊕Q[t2 ] → {α(−1)q y q +βxq } ∈ Q[x, y], and hence it is injective. 2 6. The proof of Theorem 1 By the property (2) the family P of all proper closed subgroups of T2 coincides with ∪{Ii | i ∈ N}. For ˇ q (WI , Q), Iq H ˇ q (W N , Q), N > 1, and simplicity sake, we introduce the following notations: Iq1,i H i N ∪ Ii
i=1
ˇ q (WD , Q). Then the locked cohomology H ˇ q (WP , Q) of the torus coincides with the inverse limit D H q q lim{IN , fN }, where the operator q
←
q ˇ q (W N fN : IqN = H
∪ Ii
ˇ q (WN −1 , Q) , Q) → IqN−1 = H ∪ Ii
i=1
is induced by the inclusion WN −1
∪ Ii
⊂WN
∪ Ii
i=1
.
i=1
i=1
⎧ ⎪ ⎪ ⎨Q, if q = 0 q Lemma 6.1. I2 = 0, if q = 1 or q = 2k, k ∈ N ⎪ ⎪ ⎩Q q−1 2 −1 , if q = 2k + 1, k ∈ N. Proof of Lemma 6.1. Since by Lemma 4.1 WI1 ∩ WI2 = WD , the Mayer–Vietoris sequence for WI1 and WI2 (see [6]) is as follows: e2k −e2k
δ
δ
2 1 2k 2k ... − → D2k−1 − → I2k → I2k − → I2k+1 → I2k+1 − ⊕ I2k+1 → ..., − 2 − 1,1 ⊕ I1,2 −−−−−→ D 2 1,1 1,2
2k where e2k 2 −e1 is the difference of the operators induced by the corresponding embeddings. Theorem 7 implies that this difference is an injective operator for all k > 0. In view of dim D0 = 1 and Theorem 6 this difference for k = 0 is an epimorphism. 2k+1 = 0. Hence the With help of Theorem 5 we know that D2k = Qk+1 , D2k+1 = 0, I2k 1,i = Q and I1,i Mayer–Vietoris sequence splits into a series of short exact sequences: if k = 0, then epi
δ
δ
... − →0− → I02 − → Q ⊕ Q −−→ Q − → I12 − →0− → ..., and if k > 0, then δ
mono
δ
... − →0− → I2k → Q ⊕ Q −−−−→ Qk+1 − → I2k+1 →0− − → ... 2 − 2 As a result, we obtain the desired cohomology Iq2 .
2
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⎧ ⎪ ⎪ ⎨Q, if q = 0 q Theorem 8. Given N > 1, IN = 0, if q = 1 or q = 2k, k ∈ N ⎪ ⎪ ⎩Q(N −1) q−1 2 −1 , if q = 2k + 1, k ∈ N. Proof of Theorem 8. will be performed by induction on N . The basis is valid by Lemma 6.1. We assume that for all numbers less than N, N > 2, the theorem holds, and show that for N it remains valid. Since by Lemma 4.1 WN −1 ∩ WIN = D, the Mayer–Vietoris sequence for W N = WN −1 ∪ WIN ∪ Ii
∪ Ii
i=1
i=1
∪ Ii
i=1
looks like as follows: e2k −j
δ
f 2k+1 ⊕0
δ
2k+1 2k ... − → D2k−1 − → I2k → I2k −2−−→ D2k − → I2k+1 −−N−−−−→ I2k+1 → D2k+1 − → .... N − N−1 ⊕ I1,N − N N−1 ⊕ I1,N − 2k 2k Here the operator e2k is a linear operator induced by WD → 2 is injective by Theorem 6, j : IN−1 → D 2k+1 2k+1 2k+1 is a linear operator from IN to IN−1 induced by WN −1 → W N . Since D2k+1 = 0, WN −1 , and fN ∪ Ii
∪ Ii
i=1
i=1
∪ Ii
i=1
this sequence can be written as follows: epi
δ
δ
... − →0− → I0N − → Q ⊕ Q −−→ Q − → I1N − →0− → ..., for k = 0, and f 2k+1
... − →0− → I2k → 0 ⊕ Q −−−−→ Qk+1 − → I2k+1 −−N−−→ Q(N −2)k−1 ⊕ 0 − →0− → ... N − N δ
δ
mono
2k+1 for k > 0. Its analysis implies that I2k = Q(N −1)k−1 for k > 0, and I0N = Q and I1N = 0 N = 0 and IN q−1 q−1 for k = 0. Since q = 2k + 1 and k = 2 , we obtain that IqN = Q(N −1) 2 −1 for odd q > 0. Theorem 8 is proved. 2 2k+1 Corollary 2. Each operator fN : I2k+1 → I2k+1 N N−1 is epimorphism. q ˇ q (WP , Q) and all components of this inverse spectrum are known (in particular, }=H Since lim{IqN , fN ←
q each fN is by Corollary 2 an epimorphism for q = 2k + 1, k > 0), the inverse limit of spectrum fq
fq
fq
fq
q fN +1
N lim{Iq2 ←−3− Iq3 ←−4− Iq4 ←−5− . . . ←− − IqN ←−−− . . .} =
←
lim{Q ←
q−1 2 −1
epi
←−− Q2
q−1 2 −1
epi
←−− Q3
q−1 2 −1
epi
epi
←−− . . . ←−− Q(N −1)
q−1 2 −1
epi
←−− . . .}
is a countable power Qω of Q. Theorem 1 is completely proved. References [1] T. tom Dieck, Transformation Groups and Representation Theory, Lect. Notes Math., vol. 766, Springer-Verlag, Berlin– New York, 1979. [2] S.M. Ageev, On Palais universal G-spaces and isovariant absolute extensors, Mat. Sb. 203 (6) (2012) 3–34. [3] S. Ageev, S. Bogatyi, The Banach–Mazur compactum in the dimension two, Topology Atlas, 1997, Preprint No 291. [4] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York–London, 1972. [5] S.M. Ageev, Exp, Mat. Sb. 203 (6) (2016) 3–34. [6] W.S. Massey, Homology and Cohomology Theory, Monogr. Textb. Pure Appl. Math., vol. 46, Marcel Dekker, Inc., New York, 1978. [7] Wu Yi Hsiang, Cohomology Theory of Topological Transformation Groups, Springer-Verlag, New York, Heidelberg, Berlin, 1975.
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[8] S.A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Lond. Math. Soc. Lect. Note Ser., 1977. [9] S.M. Ageev, I.V. Usimov, The cohomology ring of subspace of universal S 1 -space with finite orbit types, Topol. Appl. 160 (11) (2013) 1255–1260. [10] A.T. Fomenko, D.B. Fuchs, Homotopical Topology, Graduate Texts in Mathematics, 2016, http://www.springer.com/ series/136. [11] I.M. James, C.A. McGibbon, Phantom Maps//Handbook of Algebraic Topology, Elsever, 1995.