Higher Dimensional Knaster Continua

A generalization of Knaster Continua to higher dimensions. James E. Keesling and Vincent A Ssembatya Abstract. We generalize the definition of Knaster continua to higher dimensions and characterize analogues of ”end points” to these dimensions. These points turn out to be vertices of immersed infinite cones over real projective spaces and are homologically distinguishable in dimension higher than 2. For a given generalized Knaster continuum, there are at most 2n such points. Finally, we show that a homeomorphism on any such continuum of odd dimension lifts to exactly two homeomorphisms on the solenoid corresponding to that continuum. We give some applications of these results and suggest further research.

1. Introduction We define analogues of Knaster continua in higher dimension, as certain quotient spaces. Let A be a finite dimensional compact connected abelian topological group. The quotient space is obtained from A by identifying each point with its inverse. Call the quotient K and the quotient map π : A → K. If a point x0 ∈ A is its own inverse, then in all dimensions 6= 2, π(x0 ) is homologically distinguishable in K and therefore must be mapped to another such point under any homeomophism of K. We determine that the number of such points depends on the dimension of the continuum and the bonding map and is bounded by 2n , where n is the dimension of the continuum. This is consistent with the one dimensional case in which Knaster continua have at most two end points. In the one dimensional case, Knaster continua are obtained from solenoids by such identifications. In this case a given Knaster continuum is associated with an infinite sequence of prime numbers. The number of 2’s in the sequence of primes for a given Knaster continuum determines the number of endpoints. If there are infinitely many 2’s in the associated sequence, the Knaster continuum has one end point. There are two end points if the number of 2’s in the associated sequence if finite. In the higher dimensional case, every continuum is associated with an infinite sequence of invertible matrices with integer entries. In the case when the defining matrices are all diagonal with prime numbers as entries, the number of 2’s on the leading diagonals of these matrices determine the number of the homologically distinguishable points in K. 1991 Mathematics Subject Classification. Primary 54H25, 54F15; Secondary54F11. Key words and phrases. Knaster Continua, Higher dimensional continua, Solenoids. 1

2

JAMES E. KEESLING AND VINCENT A SSEMBATYA

In the even dimensions, we show that a homeomorphism on K lifts to two homeomorphisms on the A. In this paper, a solenoid is a finite dimensional compact connected abelian topological group. 2. Preliminaries. 1

Let S be the unit circle in the complex plane also realizable as the quotient R/Z, where R is the additive group of real numbers and Z the subgroup of the integers. Let p : R −→S 1 be the quotient map given by t 7−→ exp(2πit). Let n Q Tn = S 1 be the n-torus with the product topology from S 1 . Let pn : Rn −→ Tn i=1

be the covering map that is the product of p i.e. pn (t1 , t2 , . . . , tn ) = (p (t1 ) , p (t2 ) , . . . p (tn )) . Observe that if f : Tn −→ Tn is a homomorphism, there is a unique homomorphism on Rn represented by an n × n matrix M with integer entries satisfying f ◦ pn = pn ◦ M. 2.1. Generalized Solenoids. Let µ = (M1 , M2 , . . . ) be an infinite sequence of n × n invertible matrices with integer co-efficients. Define the µ-adic generalized solenoid Aµ to be the inverse limit space of mappings fMj , on the torus Tn . We write ∞ n Aµ = ← lim − T , fMj = {(xk )k=1 : xk = fMk (xk+1 ), k ≥ 1} where fMj are maps induced on Tn by the linear transformations on Rn whose ∞ matrices are {Mj }j=1 . M

M

M

fM

fM

fM

Rn ←−−1−− Rn ←−−2−− Rn ←−−3−− . . .       pn y pn y pn y 1 2 3 T n ←−−− − T n ←−−− − T n ←−−− − ... ∞ Q Tn , Aµ is a connected Equipped by the relativitized product topology from

i=1

compact Abelian metric topological group. Conversely, every such compact connected finite dimensional Abelian topological group is the inverse limit of tori of the same dimension. In the case when n = 1, we get the well known 1- dimensional solenoids. Let p¯n : Rn −→ Aµ be the map induced by the projections pn : Rn −→ Tn , i.e   !−1 ∞ k−1 Y p¯n (t) = p  Mi · t ¯ ¯ i=0

k=1

where t is the transpose of the vector (t1 , t2 , . . . , tn ) . ¯ We denote the image of p¯n by Γn and we may drop the indices if the dimension n is well understood. All other composants in Aµ are cosets of Aµ by the subgroup Γn . We shall denote the composant of an element x ∈ Aµ by Γ(x). We shall assume that p¯n is a 1-1 or equavalently that the dual of Aµ has no Z-factor. In this case, p¯n is a continuous homomorphism of Rn onto a dense subgroup Γµ of Aµ . This subgroup Γµ is also the arc component of the identity element in Aµ (see section

KNASTER

3

5 of [8]). The locally arcwise connected topology suggested in ([5]) realizes Γµ as Rn . The subgroup Λµ := {z = (zk )∞ k=0 ∈ Aµ : z0 = 1} of Aµ is homeomorphic to the ¯ cantor set (see [8], Lemma 5.1). It follows from Lemma 5.4 [8] that π ¯ : Λµ × Rn −→ Aµ defined by π ¯ (w, t) = p¯n (t) · w is a local homeomorphism of Λµ × Rn onto Aµ . ¯ Let dΛµ be an invariant metric on Λµ (see [6]). This induces invariant metrics on n Λµ × R and Aµ given by dΛµ ×Rn (w, t) , w, ˜ , t − t˜ ˜ t˜ := max dΛµ (w, w)

(2.1) and (2.2)

˜ t˜ : π ¯ (w, t) = z, π ¯ w, ˜ t˜ = z˜ dAµ (z, z˜) := min dΛµ ×Rn (w, t) , w,

where ||·|| is the euclidean norm on Rn . These metrics make π into a local isometry. 2.2. Knaster continua from generalized solenoids. Let µ = (M1 , M2 , . . . ) be a sequence of n × n invertible matrices with integer co-efficients. Define the µ−adic Knaster continuum Kµ to be the continuum obtained by identifying the element x ∈ Aµ with it’s inverse. So we have map from Aµ to Kµ that is at most 2-1. Let π : Aµ → Kµ be the quotient map. Define a metric (2.3) dKµ ([z] , [˜ z ]) = min dAµ (z, z˜) , dAµ z −1 , z˜ 2.3. Examples. Let µ = (M1 , M2 , . . . ) be a sequence of matrices with integer entries and let Kµ be the µ-adic Knaster continuum. An element x of Aµ is an idempotent in Aµ iff x2 = x. Each of the equivalence classes in Kµ of such elements has a single element in them. These elements come from idempotents in the Torus Tn ; There are 2n such elements in the torus: (1, 1, 1, . . . , 1), (−1, 1, 1, . . . , 1), (1, −1, 1, . . . , 1), . . . , (−1, −1, . . . , −1), and these are covered by the 2n elements: 1 1 1 1 1 (0, 0, 0, . . . , 0), ( , 0, 0, . . . , 0), (0, , 0, . . . , 0), . . . , ( , , . . . , ) 2 2 2 2 2 in Rn . The sequence of matrices µ determines which of these points lead to idempotents in the solenoid. For instance, in the case of a single matrix with diagonal prime number entries, one has the following theorem. Proposition 1. Let



p1 0  M =.  .. 0

0 p2 .. .

... ... .. .

0

...

 0 0  ..  .

pn

and let AM = Aµ , with µ = (M, M, . . . ), then the number of idempotents in Aµ is 2r where r is the number of primes on the diagonal of M that are distinct from 2.

4


Each idempotent in the solenoid leads to such a point in the Knaster continuum. We shall use homology to distinguish these points from the rest of the points in Kµ for dimension n ≥ 3. We observe that this analysis does not yield results in dimension 2. To see how this analysis would work, one needs to observe that composants in the solenoid are immersions of Rn in Aµ . In addition, if a point x ∈ Aµ is an idempotent, then all points in its composant have their inverses in that composant (this follows from the continuity of the map ψ : w 7→ w−1 on Aµ ). It follows that the idempotents are the images of the origin under the map φx ◦ p¯ that maps Rn to the composant Γ(x) of x in the solenoid. Here φx := w 7→ x · w. Also observe that inverses in Rn lie on spheres as antipodal points. So identifying inverses in the solenoid corresponds to identifying antipodal points of spheres centered at the origin in Rn . Each of these spheres then becomes a projective plane. The resulting space is an infinite cone over projective space with the idempotents as the the vertices. Let Pn be the space obtained from Rn by identifying t with −t and let p : ¯ ¯ n R → Pn be the quotient map. For a point z ∈ Kµ let C(z) be the composant of z in Kµ . We note that if z ∈ Sµ is an idempotent, then C(z) is obtained from Γ(z) by identifying points and their inverses. Theorem 1. Consider p¯ : Rn → Γ, (as earlier defined). For a given idempotent z ∈ Sµ define φz : Sµ 7→ Sµ by φz (w) = w · z. The the following diagram φz ◦p¯

Rn −−−−→ Γ(z)     py π|Γ(z) y Ψ

P −−−−→ C(z) commutes. The following Lemmas follow from the Homology exact sequences of pairs (see [10]) Lemma 1. Let z be an idempotent in Sµ , µ a sequence of diagonal matrices with integer entries. Let C(z) be the composant of [z] ∈ Kµ . Then Hn (C(z), C(z) − z) = Hn (RP n−1 ). where Hn (X, A) is the homology of the space X relative to a subset A. Lemma 2. Let z be an idempotent in Sµ , µ a sequence of diagonal matrices with integer entries. Let C(z) be the composant of [z] ∈ Kµ . Suppose x ∈ C(z), x 6= z, then Hn (C(z), C(z) − x) = Hn (S n−1 ). Lemma 3. Let z be an element in Sµ , z not an idempotent in Sµ , µ a sequence of diagonal matrices with integer entries. Let C(z) be the composant of [z] ∈ Kµ . Suppose x ∈ C(z), then Hn (C(z), C(z) − x) = Hn (S n−1 ).

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5

3. Lifting homeomorphisms from Knaster continua to solenoids Theorem 2. Let µ be a sequence of n × n invertible matrices with integer entries, n an odd potive integer, and assume the induced homomorphism p¯ : Rn → Γ is 1 − 1. Let h : Kµ → Kµ be a homeomorphism that fixes any of the vertex elements. Then there exist exactly ˜1, h ˜ 2 : Sµ → Sµ such that h ˜ i ◦ π = π ◦ h, i = 1, 2 two distinct homeomorphisms h Lemma 4. Let τ : Rn → Rn (n an odd positive integer) be a homeomorphism that fixes the origin and is isotopic to the identity. Let ρ > > 0 be positive numbers, then there exists a t with ktk, kτ (t)k ∈ [0, ρ] such that kt + τ (t)k = . Proof. Consider a sphere S centered at the origin and of radius . Suppose for every t on S the angle between the vector t and τ (t) is bigger than π2 . Define g : S n → S n by g(x) =

τ (x) kτ (x)k

and let f : S n → S n be the antipodal map x 7→ −x. Then g is homotopic to f for if H : Rn × I → R is the isotopy between τ and the identity map, then H(x, t) ˜ H(x, t) := kH(x, t)k will be a homotopy between the identity and g. This is a contradiction since S n is an even dimensional sphere. The identity is not homotopic to the antipodal map in even dimension. So there is t on S such that the angle between t and τ (t) is less than or equal to π2 and therefore kt + τ (t)k > By continuity of t + τ (t) and τ (0) = 0, there is a t such that kt + τ (t)k = . Lemma 5. There is a ρ > 0 such that if < ρ, and τ : Rn → Rn (n an odd positive integer, τ (0) = 0) is a homeomorphism isotopic to the identity and satisfying dKµ (π(s(t)z), π(s(−τ (t))z) < , t ∈ Rn , then dSµ (z, z −1 ) < 3. Proof. Since the projection Λ × Rn → Sµ is a local isometry, there exists a ρ > 0 such that dSµ (s(t)z, s(−τ (t))z) = kt+τ (t)k for t satisfying ktk, kτ (t)k ∈ [0, ρ]. By hypothesis dKµ (π(s(t)z), π(s(−τ (t))z)) = min dAµ (s(t)z, s(−τ (t))z), dAµ (s(t)z, s(+τ (t))z −1 ) < < ρ. (3.1) It follows from τ being isotopic to the identity (lemma 4) that there is a t satisfying ktk, kτ (t)k ∈ [0, ρ], kt − τ (t)k < and kt + τ (t)k = (t is chosen so that the angle between t and τ (t) is less than π/2).

6


It follows from (4.1) for that specific t that dAµ (s(t)z, s(τ (t))z −1 ) < and therefore dAµ (z, z −1) ≤ dAµ (z, s(t)z) + dAµ (s(t)z, s(τ (t)z −1 )) + dAµ (s(τ (t)z −1 ), z −1 ) < ktk + + kτ (t)k < 3 n

n

0

0

−1

Proof. Proof of Theorem 3 Let P = R ∼ where t ∼ t iff t = t and let ¯ := k ◦ h ◦ k −1 . Since n ≥ 3,˜ k : Pn → Kµ be the immersion of Pn into Kµ . Let h n n R = R \ {0} is a universal cover for˜ Pn = Pn \ {k(0)} and therefore there is a homeomorphism ¯ h : Rn → Rn which ¯ lifts h (first lift to˜ ¯ ◦ s−1 . ˜ : Γ → Γ by h ˜ := s ◦ h Rn and then extend to the origin). Define h ˜ We shall show that h is uniformly continuous. Fix > 0 such that 3 < ρ, there is a δ > 0 such that dKµ (h(x), h(y)) < 8 whenever dKµ (x, y) < δ, x, y ∈ Kµ . ˜ ˜ Suppose that dAµ (s(x1 ), s(x2 )) < δ. We shall show that dAµ (h(s(x 1 )), h(s(x2 ))) < . Because translations are isometries of Aµ , dAµ (s(x1 + t), s(x2 + t)) < δ for all t. ¯ ) So h ◦ k ([xi + t]) = k ([yi + τi (t)]), yi = h(x i n n n τi : R → R homeomorphisms of R , i = 1, 2. By the choice of δ, , ∀t. 8 If dAµ (s(y1 ), s(y2 )) < 8 the proof is complete, otherwise dAµ (s(y1 ), s(−y2 )) < n 8 . It follows that dAµ (s(y1 + t), s(−y2 + t)) < 8 for all t ∈ R . This implies that n dKµ (π(s(y1 + t)), π(s(−y2 + t))) < 8 for all t ∈ R . By the triangular inequality and τ := τ2 ◦ τ1−1 ,

(3.2)

dAµ (k([y1 + τ1 (t)])), k([y2 + τ2 (t)])