Zbigniew Błaszczyk. Properly discontinuous actions on homotopy surfaces. Uniwersytet M. Kopernika w Toruniu
|
|
- Helena Kurowska
- 5 lat temu
- Przeglądów:
Transkrypt
1 Zbigniew Błaszczyk Uniwersytet M. Kopernika w Toruniu Properly discontinuous actions on homotopy surfaces Praca semestralna nr 3 (semestr letni 2010/11) Opiekun pracy: Marek Golasiński
2 Term paper #3: Properly discontinuous actions on homotopy surfaces Zbigniew Bªaszczyk Abstract Fujii has observed that a nite group G acts freely on a surface M other than the 2-sphere or the projective plane if and only if there exists a group extension 1 π 1(M) π 1(N) G 1, where N is a surface such that χ(m) = G χ(n). We discuss an extension of this result to the case of properly discontinuous and cellular actions of arbitrary groups on homotopy surfaces, with the key ingredient in our approach being the notion of cohomological dimension of a group. Among applications, we prove that the family of nite groups which act freely on a surface M is precisely the same as the family of nite groups which act freely and cellularly on homotopy M's. We also deduce that a torsionfree group with innite cohomological dimension cannot act properly discontinuously and cellularly on any homotopy surface, with four possible exceptions. Special attention is given to M 2, the orientable surface of genus 2. In particular, we provide a complete list of groups which act properly discontinuously on M 2 R. 1 Introduction 1.1. Given a closed manifold M, a question that is often of interest is the following: which nite groups act freely on M? One can also be inclined to consider its `homotopic' counterpart: which nite groups act freely and cellularly on nite CW-complexes homotopy equivalent to M? Looking simultaneously at both versions of the question proved to be very fruitful; the solution of the spherical space form problem springs to mind (see, for example, [5]). Having settled the issue for nite groups, a more general problem emerges, namely that of providing a description of groups which act properly discontinuously on M R m, or, more generally, on nite-dimensional CW-complexes homotopy equivalent to M. This sort of question has been studied especially extensively in the case when M is a sphere (see [4], [10], [11], [17], [18]). Since Date: June 23, 2011; Revised: November 4, Mathematics Subject Classication: Primary 57S25, Secondary 20J06 Key words and phrases: (virtual) cohomological dimension, group extension, homotopy surface, properly discontinuous action, surface group 1
3 spheres are the simplest manifolds homologically, it seems only natural to consider a similar problem for the nicest possible spaces in the sense of homotopy, i.e., for aspherical manifolds. With this idea in mind, we investigate properly discontinuous and cellular transformation groups of nite-dimensional CWcomplexes homotopy equivalent to a surface M, for M other than the 2-sphere or the projective plane. Considering that the homotopy type of an aspherical CW-complex is determined by its fundamental group, it is perhaps not surprising that this problem can be reduced to a question about the existence of a specic group extension Recall the following theorem due to Fujii. Theorem 1.1 ([8, Theorem 1.6]). A nite group G acts freely on a surface M other than the 2-sphere S 2 or the projective plane RP 2 if and only if there exists a group extension 1 π 1 (M) π 1 (N) G 1, where N is a surface such that χ(m) = G χ(n). This result gives a reasonably ecient method of determining which nite groups act freely a surface M, especially if M is of low genus. For example, Fujii has been able to show that a nite group acts freely on the Klein bottle if and only if it cyclic of order either 2m + 1 or 4m + 2, m 0 ([8, Theorem 1.8]). We discuss a generalization of this result to the case of properly discontinuous and cellular actions of arbitrary groups on homotopy surfaces, with the key ingredient in our approach being the notion of cohomological dimension for groups (Proposition 3.1). This turns out to have some interesting consequences. It allows us to show that the family of nite groups which act freely on a surface M is precisely the same as the family of nite groups which act freely and cellularly on homotopy M's (Corollary 3.4). We also derive a lower bound on the dimension of a homotopy surface which admits a properly discontinuous and cellular action of a group with nite virtual cohomological dimension; this extends results obtained by Golasi«skiGonçalvesJimenéz [11, Proposition 3.4] and Lee [15, Theorem 5.2] (Theorem 4.2). As another application, we prove that a torsionfree group with innite cohomological dimension cannot act properly discontinuously and cellularly on any homotopy M, with possible exceptions when M is either the 2-sphere, the torus, the projective plane, or the Klein bottle (Theorem 5.2). One notable feature of our approach is that it makes the task of showing that certain groups act properly discontinuously and cellularly on homotopy surfaces particularly easy. Most of Section 6 is dedicated to exhibiting this sentiment. For example, we show that the general linear group GL(n, Z) acts in such a way on homotopy orientable surfaces of genus (p n 1)(p n p) (p n p n 1 ) + 1, where p 3 is a prime number (Corollary 6.2). Special attention is given to M 2, the orientable surface of genus 2. In particular, we determine all groups that act properly discontinuously on M 2 R (Proposition 7.1). In fact, there are only four of them: the cyclic group of order 2, the innite cyclic group, the direct product Z Z 2 and the semidirect product Z Z Throughout this paper, the term surface is taken to mean a closed and connected 2-dimensional manifold. 2
4 Since there does not seem to be a generally accepted denition of the term `properly discontinuous action', we shall state explicitly what we mean by this. An action G X X is said to be properly discontinuous if for any x X there exists its neighbourhood U such that gu U = for any g G, g 1. The bottom line is that we want the natural projection X X/G to be a covering. Apart from these, we employ the standard notation Quite a few results presented during the course of this paper can be promoted to higher-dimensional aspherical manifolds. The emphasis, however, is not on the greatest possible generality, but rather on specic calculations regarding homotopy surfaces. 2 Cohomological dimension 2.1. Since our approach relies heavily on the notion and properties of cohomological dimension, we will take some time to recall the related facts. The cohomological dimension of a group Γ, written cd Γ, is dened to be the projective dimension of Z over the integral group ring ZΓ. It is not hard to see that cd Γ = inf{n H i (Γ, ) = 0 for i > n} = sup{n H n (Γ, M) 0 for a Γ-module M}, where H (Γ, ) stands for the cohomology group of Γ. The topological analogue of cd Γ, the geometric dimension of Γ (gd Γ), is dened to be the minimal dimension of a classifying space for Γ. By results of EilenbergGanea [6], Stallings [20] and Swan [21], cd Γ = gd Γ whenever cd Γ 2. In the case when cd Γ = 2, it is still true that gd Γ 3. It is well-known that a group Γ with torsion necessarily has cd Γ =. Because of this the following notion has been introduced: a group Γ is said to have nite virtual cohomological dimension if there exists a subgroup Γ Γ with cd Γ < and of nite index; we then set vcd Γ = cd Γ. By Serre's theorem [3, Chapter VII, Theorem 3.1], vcd Γ does not depend on the choice of Γ. Observe that if vcd Γ <, we may choose Γ Γ to be normal; it suces to consider the normal core of any such Γ. We will repeatedly make use of the following facts: (1) If Γ is a virtually torsionfree group and Γ Γ is any subgroup, then vcd Γ vcd Γ, with equality if the index [Γ : Γ ] is nite. (2) Given a group extension 1 Γ Γ Γ 1 with cd Γ < and vcd Γ <, the inequality vcd Γ cd Γ + vcd Γ holds. Both of these properties are best proved by looking at the torsionfree case rst: the former then follows from the fact that a projective resolution of Z over ZΓ can be regarded as a projective resolution of Z over ZΓ, the latter is a straightforward consequence of the LyndonHochschildSerre spectral sequence Recall that any group extension 1 Γ Γ Γ 1 gives rise to a homomorphism Γ Out(Γ ), where Out stands for the outer automorphism group; consult [3, Chapter IV, Section 6] for details. What we need is a sort of converse: 3
5 Proposition 2.1 ([3, Chapter IV, Corollary 6.8]). If Γ has trivial center then there is exactly one (up to equivalence) extension of Γ by Γ corresponding to any homomorphism ϕ: Γ Out(Γ ). In the above situation, the unique extension of Γ by Γ can be described as the bered product of the diagram Γ ϕ Aut(Γ ) Out(Γ ). 3 A generalization of Fujii's theorem Let M n be an n-dimensional manifold. A homotopy manifold of type M n (or a homotopy M n ) is a nite-dimensional CW-complex X homotopy equivalent to M n. Whenever n = 2 and M is closed and connected, we say that X is a homotopy surface. Our rst goal is to obtain a version of Theorem 1.1 for homotopy surfaces; this is achieved in Proposition 3.1 below. As we have already mentioned, the key ingredient in our approach is the notion of cohomological dimension. In what follows, the fundamental group of a surface is frequently called a surface group. Proposition 3.1. (1) A group G acts properly discontinuously and cellularly on a homotopy surface X of type M S 2, RP 2 if and only if there exists a group extension 1 π 1 (M) Γ G 1, where Γ is a group with cd Γ <. In this case, the orbit space X/G is a nite-dimensional classifying space for the group Γ. (2) In particular, a nite group G acts freely and cellularly on a homotopy surface of type M S 2, RP 2 if and only if there exists a group extension 1 π 1 (M) π 1 (N) G 1, where N is a surface such that χ(m) = G χ(n). Proof. (1) Suppose a group G acts properly discontinuously and cellularly on a homotopy surface X of type M S 2, RP 2. Consider the bration G X X/G; inspection of its long exact sequence of homotopy groups reveals that the orbit space X/G is aspherical and that π 1 (X/G) ts into the short exact sequence 1 π 1 (M) π 1 (X/G) G 1. Moreover, since X/G is a nite-dimensional classifying space for π 1 (X/G), it follows that cd π 1 (X/G) <. Conversely, consider a group extension 1 π 1 (M) Γ G 1, 4
6 where Γ is a group with cd Γ <. Let X be a nite-dimensional classifying space for Γ and let X be the covering space of X corresponding to the subgroup π 1 (M) Γ. Then G = Γ/π 1 (M) acts properly discontinuously and cellularly on X. Since X is a nite-dimensional aspherical CW-complex with π 1 ( X) = π 1 (M), we have X M and the conclusion follows. (2) Proceed along the same lines as above. For the `only if' part, recall that a group with nite cd is torsionfree. By a theorem of Zieschang [23], [24], a torsionfree extension of a nite group by a surface group is again a surface group, hence π 1 (X/G) = π 1 (N) for a surface N. The asserted equality is a consequence of the general behaviour of χ; see [3, Chapter IX, Section 6]. Remark 3.2. (1) The above proof in fact yields one more piece of information: if gd Γ = n, then G acts properly discontinuously and cellularly on an n-dimensional homotopy surface of type M. (2) Note that the assumption of cellularity of the action is imposed in order to ensure that the orbit space is a nite-dimensional CW-complex. In the case when the homotopy surface in question is a manifold itself, the orbit space under a properly discontinuous action is again a manifold, thus cellularity can be neglected. More often than not we will state a result only in the cellular version, understanding that it can be carried over to the geometric case. Example 3.3. Given a surface M S 2, RP 2 and a group G with cd G <, there exists a homotopy surface of type M which admits a properly discontinuous and cellular G-action. To see this, set Γ = π 1 (M) G in Proposition 3.1; since both π 1 (M) and G have nite cohomological dimension, this is also the case for Γ. The geometric counterpart of this phenomenon can be obtained by taking the product EG M, EG being the universal cover of a nite-dimensional classifying space for G, with the usual G-action on the rst summand and the trivial one on the second. Let us record some immediate consequences of Proposition 3.1. Corollary 3.4. (1) A nite group acts freely and cellularly on a homotopy surface of type M if and only if it acts freely on M. (2) A nite group acts freely on a surface M if and only if it acts freely and cellularly on M. Proof. (1) Combine Theorem 1.1 with the second part of Proposition 3.1. (2) The `if' part is trivial. For the `only if' part, apply Theorem 1.1 to obtain an appropriate group extension. Now proceed as in the proof of Proposition 3.1, with the following improvement: take X to be the surface N itself. Then X is again a surface and, since surfaces are classied by their fundamental groups up to homeomorphism, X M. Recall that a group Γ is said to be of type FL if Z admits a nite free resolution over ZΓ. If, additionally, Γ is nitely presented, this property amounts to the existence of a nite classifying space for Γ ([3, Chapter VIII, Theorem 7.1]). Corollary 3.5. Suppose a group G acts properly discontinuously and cellularly on a homotopy surface X. 5
7 (1) The orbit space X/G is a homotopy surface if and only if G is nite. (2) If X/G has the homotopy type of a nite CW-complex, then G is nitely generated. Conversely, if G is nitely presented and of type F L, then X/G has the homotopy type of a nite CW-complex. Proof. (1) This is a straightforward consequence of Proposition 3.1 and the subgroup structure of surface groups: any subgroup of nite index of a surface group is again a surface group, while any subgroup of innite index is free. See, for example, [1, Theorem 2.1]. (2) If X/G has the homotopy type of a nite CW-complex, then π 1 (X/G) is nitely generated. Due to the existence of an epimorphism π 1 (X/G) G, so is G. The key observation for the converse is that π 1 (X/G) is nitely presented and of type F L. This holds because both π 1 (M) and G are so, and these two properties are well-known to be closed under extension. By the preceding remark, X/G has the homotopy type of a nite CW-complex. 4 The dimension of a homotopy surface A natural question to consider is the following: assuming a group G acts properly discontinuously and cellularly on a homotopy surface X of a given type, what is the minimal dimension of X as a CW-complex? Clearly, dim X 2, but it is reasonable to expect a better bound in terms of cohomological dimension of G, vide [11, Proposition 3.4] or [15, Theorem 5.2]. This is indeed the case, at least for homotopy surfaces of an orientable type, and we now set out to make this assertion precise. Proposition 4.1. Let M = M n be an aspherical, closed, orientable manifold and G a group with vcd G = m <. Given an extension 1 π 1 (M) Γ G 1, the equality vcd Γ = cd π 1 (M) + vcd G holds. Proof. We can assume without loss of generality that G has nite cohomological dimension, for otherwise we can pass to its torsionfree subgroup of nite index. Write π for π 1 (M). Choose a G-module A such that H m (G, A) 0 and consider the LyndonHochschildSerre spectral sequence of the above extension: E p,q 2 = H p( G, H q (π, A) ) H p+q (Γ, A). By the usual spectral sequence argument, H m+n (Γ, A) = H m( G, H n (π, A) ). The Universal Coecient Theorem gives an extension of G-modules 0 Ext ( H n 1 (π, Z), A ) H n (π, A) Hom ( H n (π, Z), A ) 0. Now apply the associated long exact sequence of cohomology groups to obtain a surjection H m( G, H n (π, A) ) H m (G, A). (We can always assume the induced action G Aut ( H n (π, Z) ) is trivial, hence there is an isomorphism Hom ( H n (π, Z), A ) = A of G-modules.) Consequently, H m+n (Γ, A) 0. We are now ready to consider an extension of both [11, Proposition 3.4] and [15, Theorem 5.2]. 6
8 Theorem 4.2. Let M = M n be an aspherical, closed and orientable manifold and G a group with vcd G <. (1) If G acts properly discontinuously and cellularly on a homotopy manifold X of type M, then dim X vcd G + n. (2) If G acts properly discontinuously on M R m, then vcd G m, with equality if and only if M R m /G is closed. Proof. (1) Just as in the proof of Proposition 3.1, the orbit space X/G is a nite-dimensional aspherical CW-complex and its fundamental group is given by the extension 1 π 1 (M) π 1 (X/G) G 1. By Proposition 4.1, cd π 1 (X/G) = cd π 1 (M) + cd G. Furthermore, dim X = dim X/G gd π 1 (X/G) cd π 1 (X/G), and the conclusion follows. (2) The above proof works equally well for the rst part of (2), only that now M R m /G is an aspherical manifold and dim stands for the dimension of a space as a manifold. For the second part, recall that whenever a space Y is simultaneously a classifying space for a group Γ and a d-dimensional manifold, then cd Γ d, with equality if and only if Y is closed ([3, Chapter VIII, Proposition 8.1]). This and a similar string of (in)equalities as above imply the claimed result. Clearly, vcd Γ = 0 if and only if Γ is a nite group. By results of Stallings [20] and Swan [21], vcd Γ = 1 if and only if Γ is a virtually free group. Consequently, we have: Corollary 4.3. Suppose a group G with vcd G < acts properly discontinuously and cellularly on a homotopy surface X of an orientable type. (1) If dim X = 2, then G is nite. (2) If dim X = 3, then G is either nite or virtually free. 5 The non-existence of actions of torsionfree groups with innite cd We now turn attention to the problem of determining which groups act properly discontinuously and cellularly on homotopy surfaces. Example 3.3 shows that we need not worry about groups with nite cohomological dimension, as this case is trivial in the sense that `everything acts on everything'. This leaves us in the realm of groups with innite cohomological dimension, and the rst question that springs to mind is whether a torsionfree group with innite cohomological dimension can act properly discontinuously and cellularly on a homotopy surface. We answer this question in the negative in Theorem 5.2, albeit with four possible exceptions. Proposition 5.1. The outer automorphism group of the fundamental group of any surface has nite virtual cohomological dimension. 7
9 Proof. It is well-known that the outer automorphism group of a surface group is the extended mapping class group of that surface. (For orientable surfaces, this is the celebrated DehnNielsenBaer Theorem.) (Orientable surfaces) Harer [13, Theorem 4.1] has shown that the mapping class group of an orientable surface has nite virtual cohomological dimension. Since the mapping class group of an orientable surface M is an index 2 subgroup of the extended mapping class group of M, the conclusion follows. (Nonorientable surfaces) A result of HopeTillmann [14, Lemma 2.1] states that the mapping class group of a nonorientable surface of genus g is a subgroup of the mapping class group of an orientable surface of genus g 1, which gives the desired upper bound. Theorem 5.2. Let M be a surface other than the 2-sphere, the torus, the projective plane, or the Klein bottle. A torsionfree group with innite cohomological dimension cannot act properly discontinuously and cellularly on any homotopy surface of type M. Proof. Suppose a torsionfree group G with cd G = acts properly discontinuously and cellularly on a homotopy surface X of type M. Similarly as in the proof of Proposition 3.1, the bration G X X/G yields the extension 1 π 1 (M) Γ G 1, which in turn gives rise to a homomorphism ϕ: G Out ( π 1 (M) ). Since the center of π 1 (M) is trivial by a result of Griths [12, Theorem 4.4], it follows from Proposition 2.1 that Γ is uniquely determined by ϕ. More precisely, there is an isomorphism Γ = Aut ( π 1 (M) ) Out(π1(M)) G = { (σ, g) Aut ( π 1 (M) ) G η(σ) = ϕ(g) }, where η : Aut ( π 1 (M) ) Out ( π 1 (M) ) is the natural projection. The extension 1 ker ϕ G im ϕ 1 gives the inequality cd G cd ker ϕ + vcd im ϕ. Since vcd im ϕ < by Proposition 5.1 and cd G = by hypothesis, it follows that cd ker ϕ =. To conclude the proof, observe that π 1 (M) ker ϕ is a subgroup of Γ, hence cd Γ =. This yields a contradiction with Proposition 3.1. A careful examination of the proof of Theorem 5.2 allows one to claim: Corollary 5.3. If a nite group G acts freely on a homotopy surface of type M as above, then G is a subgroup of Out ( π 1 (M) ). Proof. Similarly as above, we arrive at a homomorphism ϕ: G Out ( π 1 (M) ) and an explicit description of Γ. If ker ϕ were not trivial, it would amount to torsion in Γ, thus making its cohomological dimension innite. Example 5.4. It is certainly possible, in general, for a torsionfree group of innite cohomological dimension to act properly discontinuously and cellularly on a nite-dimensional CW-complex. 8
10 (1) Let F denote a free group on a countably innite number of generators. Then cd F = 1 and, since the commutator subgroup [F, F ] F is also free, cd [F, F ] = 1. Hence we obtain the extension 1 [F, F ] F Z 0, which yields a properly discontinuous and cellular action of i=1 Z on a 1-dimensional CW-complex homotopy equivalent to a bouquet of certain number of circles. (2) The above construction shows that, in fact, any group G acts properly discontinuously and cellularly on some 1-dimensional homotopy bouquet of circles in such a way that the orbit space is again a 1-dimensional homotopy bouquet of circles. Too see this, simply take any free resolution of G of length 1. Remark 5.5. As Z n i=1 Z for any n 0, it follows from Theorem 4.2 that i=1 Z cannot act properly discontinuously and cellularly on a homotopy manifold of type M for M aspherical, closed and orientable. A spectral sequence argument similar to that of Proposition 4.1 shows that this also cannot happen when M is a nonorientable surface. There are, however, examples of torsionfree groups with innite cohomological dimension and no innitely generated free abelian subgroup. i=1 6 Actions of groups with nite vcd 6.1. Having settled the issue for torsionfree groups with innite cohomological dimension, we will now consider the behaviour of groups with torsion, especially those with nite virtual cohomological dimension. After series of very general results and examples, we concentrate on actions on homotopy surfaces of a specied type. To begin with, consider the following generalization of the phenomenon described in Example 3.3. Proposition 6.1. Let G be a group with vcd G < and N G a normal torsionfree subgroup of nite index. If G/N acts freely on a surface M, then G acts properly discontinuously and cellularly on a ( [G : N] gd G + 2 ) -dimensional homotopy surface of type M. Proof. Using Corollary 3.4, we can assume without loss of generality that G/N acts freely and cellularly on M. Then G acts cellularly, although not properly discontinuously, on M via the homomorphism G G/N Homeo (M). On the other hand, by [3, Chapter VIII, Theorem 11.1], there exists a contractible and proper 1 G-complex X of dimension [G : N] gd N. [Take X to be the space of N-equivariant maps G EN, endowed with the compact-open topology. One then easily sees that X (EN) [G:N].] Clearly, X is a free N-complex. It is routine to verify that the action G (X M) X M given by g(x, m) = ( gx, (gn)m ) 1 Recall that a G-complex X is said to be proper if the isotropy group Gx of any x X is nite and any x X has a neighbourhood U such that gu U =, g G \ G x. 9
11 is properly discontinuous and cellular. Corollary 6.2. The general linear group GL(n, Z), n 1, acts properly discontinunously and cellularly on a homotopy surface of type M gp, where M gp is the orientable surface of genus g p = (p n 1)(p n p) (p n p n 1 ) + 1. Proof. Recall that the general linear group GL(n, Z p ), where n 1 and p 3 is a prime number, acts freely on the orientable surface of genus g p. Indeed, by Waterhouse's [22], GL(n, Z p ) is generated by two elements, say A 1, A 2, thus there is a projection π 1 (M 2 ) = x 1, x 2, y 1, y 2 [x1, y 1 ][x 2, y 2 ] = 1 GL(n, Z p ) given by x i A i, y i 1, i = 1, 2. As already mentioned, the kernel of this map is the surface group π 1 (M g ) of the orientable surface M g. The equality χ(m g ) = GL(n, Z p ) χ(m 2 ) gives g = g p and the conlusion follows. It is well-known that GL(n, Z) has nite virtual cohomological dimension. Furthermore, the kernel of the projection GL(n, Z) GL(n, Z p ), the so called principal congruence subgroup, is torsionfree, hence the group GL(n, Z) acts properly discontinuously and cellularly on a homotopy surface of type M gp by Proposition 6.1. Let G be a one-relator group with torsion. Then vcd G 2, as explained in [3, Chapter XIII, Section 11, Example 3]. FischerKarrassSolitar have shown that a normal torsionfree subgroup N G of nite index is free if and only if G is the free product of a free group and a nite cyclic group ([7, Theorem 3]). As a consequence, we deduce: Corollary 6.3. Let F be a free group. If the cyclic group Z n acts freely on a surface M, then the free product F Z n acts properly discontinuously and cellularly on an (n + 2)-dimensional homotopy surface of type M. Proof. Write R for the generator of Z n F Z n. Consider the short exact sequence p 1 N F Z n Z n 0, where p is the obvious projection. Since the elements of nite order in F Z n are conjugates of powers of R (see [16, Theorem 4.13]), it follows that N is torsionfree. The preceding discussion now implies that N is a free group, hence gd N = 1. Applying Proposition 6.1 concludes the proof. We can oer the following improvement of Proposition 6.1 in the case when G is a semidirect product: Proposition 6.4. If a nite group H acts freely on a surface M other than the 2-sphere, the torus, the projective plane, or the Klein bottle, and N is a group with cd N <, then any semidirect product N H acts properly discontinuously and cellularly on a (gd N + 2)-dimensional homotopy surface of type M. Proof. As usual, the free H-action on M gives rise to the group extension 1 π 1 (M) Γ H 1, where Γ = Aut ( π1 (M) ) Out(π1(M)) H. This in turn induces a homomorphism ϕ: H Out ( π 1 (M) ). 10
12 Dene ψ : N H Out ( π 1 (M) ) by ψ(n, h) = ϕ(h) for any (n, h) N H. By Proposition 2.1, ψ gives rise to the group extension 1 π 1 (M) Γ N H 1, with Γ = Aut ( π 1 (M) ) Out(π1(M)) (N H). In order to conclude the proof it is enough to show that Γ is torsionfree, for in this case cd Γ < and Proposition 3.1 is applicable. This last assertion, however, is a straightforward consequence of the observation that ( σ, (n, h) ) Γ if and only if (σ, h) Γ. Remark 6.5. Consult [8], [9] for many examples of nite groups which act freely on specic surfaces, particularly those of low genus We now turn our attention to actions on homotopy surfaces of a specied type. Let M 2 denote the orientable surface of genus 2 and N g the nonorientable surface of genus g. Proposition 6.6. A group G acts properly discontinuously and cellularly on a homotopy surface X of type M 2 if and only if either cd G < or G = G Z 2, with cd G <. Proof. The `if' part follows from Example 3.3 and Proposition 6.4, respectively. Conversely, if G is torsionfree, then cd G < by Theorem 5.2. Assume that G has torsion and observe that if H G is a nontrivial nite subgroup, then the induced action H Aut ( H 2 (X, Z 3 ) ) is nontrivial. Indeed, by Proposition 3.1, H = Z 2 and χ(x/h) = 1. On the other hand, using the transfer isomorphism H (X/H; Z 3 ) H (X; Z 3 ) H, we obtain: χ(x/h) = 1 dim Z3 H 1 (X; Z 3 ) H + dim Z3 H 2 (X; Z 3 ) H. Thus if the action H Aut H 2 (X, Z 3 ) were trivial, it would imply the equality dim Z3 H 1 (X; Z 3 ) H = 3. A simple linear algebra argument shows that this is impossible. Consequently, we have an epimorphism G Aut H 2 (X, Z 3 ) = Z 2 with torsionfree kernel, which again by Theorem 5.2 has nite cohomological dimension. An appropriate group extension splits and the conclusion follows. By Proposition 3.1, homotopy N 3 's do not admit free and cellular actions of nite groups. Combining this with Example 3.3 and Theorem 5.2 thus yields: Proposition 6.7. A group acts properly discontinuously and cellularly on a homotopy surface of type N 3 if and only if it has nite cohomological dimension. This also gives a counterexample to the converse of Corollary 5.3: it is known that Out ( π 1 (N 3 ) ) = GL(2, Z) see, for example, [2, Theorem 3] and this group obviously contains nite subgroups. Remark 6.8. For the case of the 2-sphere, consult [10] and [15, Section 7]. The former paper deals with properly discontinuous and cellular actions on 11
13 homotopy 2n-spheres, while the latter treats properly discontinuous actions on S 2n R m specically. As for the projective plane, it is clear that homotopy RP 2 's admit properly discontinuous and cellular actions of only torsionfree groups. Furthermore, given any group G with nite cohomological dimension, there always exists a homotopy RP 2 which admits a properly discontinuous and cellular G-action. At present, however, I do not know whether this is possible for a torsionfree group with innite cohomological. In general, the higher the genus of the surface in question, the more dicult the task at hand. We can, however, oer some partial results. We will be interested in virtually cyclic groups which act properly discontinuously and cellularly on homotopy surfaces of type N p+2, where p is a prime number. Recall the following classication theorem due to Wall. Theorem 6.9 ([19, Theorem 5.12]). Virtually cyclic groups come in three types: (1) nite groups, (2) semidirect products H Z, with H nite, (3) G 1 H G 2, with [G i : H] = 2 for i = 1, 2 and H nite. Following [11], we will say that a group is of `type I', if it satises (2), and of `type II', if it satises (3). Remark It will be useful to know in the next section that a nitely generated group has two ends if and only if it is either of `type I' or of `type II'. In fact, this assertion is also a part of [19, Theorem 5.12]. By Proposition 3.1, the only (nontrivial) nite group which can act freely and cellularly on a homotopy surface of type N p+2 is the cyclic group of order p. That such an action actually exists has been proven by Fujii [9, Proposition 1.4]. Corollary 3.4 shows that this action can be taken to be cellular. Proposition (1) A nontrivial virtually cyclic group G acts properly discontinously and cellularly on a homotopy surface of type N 4 if and only if G is either cyclic of order 2, innite cyclic, the direct product Z 2 Z, or the semidirect product Z Z 2. (2) If a nontrivial virtually cyclic group G acts properly discontinuously and cellularly on a homotopy surface of type N p+2, p 3 a prime, then G is either cyclic of order p, innite cyclic, or a semidirect product Z p Z. Proof. The preceding discussion disposes of the case when G is nite. If G is of `type I', then G = H Z, where H is a nite group. By the previous case, H is either trivial or H = Z p. That both Z and Z p Z act properly discontinuously and cellularly on N p+2 R is clear. We move on to groups of `type II'. Recall that both G 1 and G 2 can be regarded as subgroups of G 1 H G 2 ; see, for example, [3, Chapter II, Lemma 7.4]. Consequently, in the rst case, G = Z 2 Z 2. It is, however, well-known that Z 2 Z 2 = Z Z2. This group acts properly discontinuously and cellularly on a homotopy surface of type N 4 by Proposition
14 Finally, in the second case, G cannot be of `type II', since Z p, p 3, does not contain an index 2 subgroup. At this point I do not know which of the semidirect products Z p Z actually act properly discontinuously and cellularly on homotopy N p+2 's. 7 Groups that act on M 2 R We hinted in the introduction that investigating transformation groups of homotopy surfaces is not only of intrinsic interest, but should also yield information regarding transformation groups of manifolds of the form M R m. To back that sentiment up, we will show that the elaborated methods allow to determine all groups which act properly discontinuously on M 2 R. Even though this is only one example, it feels like it deserves its own section: virtually every result presented during the course of this paper is used in its proof. Proposition 7.1. A nontrivial group G acts properly discontinuously on M 2 R if and only if it either cyclic of order 2, innite cyclic, the direct product Z Z 2, or the semidirect product Z Z 2. Proof. Suppose a nontrivial group G acts properly discontinuously on M 2 R. By Proposition 6.6, vcd G <, hence Theorem 4.2 is applicable and implies that vcd G 1. If vcd G = 0, then G is nite and G = Z 2 by Proposition 3.1. Since this group acts freely on M 2, obviously it acts freely also on M 2 R. Suppose that vcd G = 1. Again by Theorem 4.2, the orbit space M 2 R/G is a closed manifold. Therefore, by Corollary 3.5, G is nitely generated. The theory of ends (see, for example, [19, Section 5]) now gives e(g) = e(m 2 R) = 2. Using Wall's criterion (Remark 6.10), we see that G is either innite cyclic, the direct product Z Z 2, or the semidirect product Z Z 2. That the rst two groups act properly discontinuously on M 2 R is clear. It remains to show that this is also the case for Z Z 2. To see this, write F 2 = a, b for the free group on two generators. Dene ϕ: F 2 Homeo(M 2 R) by setting {a ( (x, t) (x, t + 1) ) b ( (x, t) (bx, t) ) for any (x, t) M 2 R, where bx comes from a free Z 2 -action on M 2. One easily sees that ϕ is trivial on the normal closure of {b 2, baba}, hence it induces a Z Z 2 -action on M 2 R. It is routine to verify that this action is in fact properly discontinuous. Acknowledgements. I have been supported by the joint PhD programme ` rodowiskowe Studia Doktoranckie z Nauk Matematycznych' during the preparation of this paper. I would like to express my gratitude to Professor Marek Golasi«ski for many invaluable comments and discussions. 13
15 References [1] P. Ackermann, B. Fine, and G. Rosenberger, On surface groups: Motivating examples in combinatorial group theory, Groups St Andrews 2005, Volume 1, London Mathematical Society Lecture Note Series 339, Cambridge University Press, 2007, pp [2] J. S. Birman and D. R. J. Chillingworth, On the homeotopy group of a nonorientable surface, Proc. Camb. Phil. Soc. 71 (1972), [3] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag, New York, [4] F. X. Connolly and S. Prassidis, On groups which act freely on R m S n 1, Topology 28 (1989), [5] J. Davis and R. J. Milgram, A Survey of the Spherical Space Form Problem, Mathematical Reports 2, Harwood Academic Publishers, [6] S. Eilenberg and T. Ganea, On the LusternikSchnirelmann category of abstract groups, Ann. of Math. 65 (1957), [7] J. Fischer, A. Karrass, and D. Solitar, On one-relator groups having elements of nite order, Proc. Amer. Math. Soc. 33 (1972), [8] K. Fujii, A note on nite groups which act freely on closed surfaces, Hiroshima Math. J. 5 (1975), [9] K. Fujii, A note on nite groups which act freely on closed surfaces II, Hiroshima Math. J. 6 (1976), [10] M. Golasi«ski, D. L. Gonçalves, and R. Jimenéz, Properly discontinuous actions of groups on homotopy 2n-spheres, preprint. [11] M. Golasi«ski, D. L. Gonçalves, and R. Jimenéz, Properly discontinuous actions of groups on homotopy circles, preprint. [12] H. B. Griffiths, The fundamental group of a surface, and a theorem of Schreier, Acta Math. 110 (1963), 117. [13] J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), [14] G. Hope and U. Tillmann, On the Farrell cohomology of the mapping class group of non-orientable surfaces, Proc. Amer. Math. Soc. 137 (2009), [15] J. B. Lee, Transformation groups on S n R m, Topology Appl. 53 (1993), [16] W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Pure and Applied Mathematics XIII, Interscience Publishers, New York, [17] G. Mislin and O. Talelli, On groups which act freely and properly on nitedimensional homotopy spheres, Computational and Geometric Aspects of Modern Algebra, London Mathematical Society Lecture Note Series 275, Cambridge University Press, 2000, pp [18] S. Prassidis, Groups with innite virtual cohomological dimension which act freely on R m S n 1, J. Pure Appl. Algebra 78 (1992), [19] P. Scott and C. T. C. Wall, Topological methods in group theory, Homological Group Theory, London Mathematical Society Lecture Note Series 36, Cambridge University Press, 1979, pp [20] J. R. Stallings, On torsionfree groups with innitely many ends, Ann. of Math. 88 (1968),
16 [21] R. G. Swan, Groups of cohomological dimension one, J. Algebra 12 (1969), [22] W. C. Waterhouse, Two generators for the general linear group over nite elds, Linear and Multilinear Algebra 24 (1989), [23] H. Zieschang, On extensions of fundamental groups of surfaces and related groups, Bull. Amer. Math. Soc. 77 (1971), [24] H. Zieschang, Addendum to: On extensions of fundamental groups of surfaces and related groups, Bull. Amer. Math. Soc. 80 (1974), Zbigniew Bªaszczyk Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/ Toru«, Poland 15
Helena Boguta, klasa 8W, rok szkolny 2018/2019
Poniższy zbiór zadań został wykonany w ramach projektu Mazowiecki program stypendialny dla uczniów szczególnie uzdolnionych - najlepsza inwestycja w człowieka w roku szkolnym 2018/2019. Składają się na
Weronika Mysliwiec, klasa 8W, rok szkolny 2018/2019
Poniższy zbiór zadań został wykonany w ramach projektu Mazowiecki program stypendialny dla uczniów szczególnie uzdolnionych - najlepsza inwestycja w człowieka w roku szkolnym 2018/2019. Tresci zadań rozwiązanych
Revenue Maximization. Sept. 25, 2018
Revenue Maximization Sept. 25, 2018 Goal So Far: Ideal Auctions Dominant-Strategy Incentive Compatible (DSIC) b i = v i is a dominant strategy u i 0 x is welfare-maximizing x and p run in polynomial time
Zakopane, plan miasta: Skala ok. 1: = City map (Polish Edition)
Zakopane, plan miasta: Skala ok. 1:15 000 = City map (Polish Edition) Click here if your download doesn"t start automatically Zakopane, plan miasta: Skala ok. 1:15 000 = City map (Polish Edition) Zakopane,
Tychy, plan miasta: Skala 1: (Polish Edition)
Tychy, plan miasta: Skala 1:20 000 (Polish Edition) Poland) Przedsiebiorstwo Geodezyjno-Kartograficzne (Katowice Click here if your download doesn"t start automatically Tychy, plan miasta: Skala 1:20 000
Convolution semigroups with linear Jacobi parameters
Convolution semigroups with linear Jacobi parameters Michael Anshelevich; Wojciech Młotkowski Texas A&M University; University of Wrocław February 14, 2011 Jacobi parameters. µ = measure with finite moments,
Machine Learning for Data Science (CS4786) Lecture11. Random Projections & Canonical Correlation Analysis
Machine Learning for Data Science (CS4786) Lecture11 5 Random Projections & Canonical Correlation Analysis The Tall, THE FAT AND THE UGLY n X d The Tall, THE FAT AND THE UGLY d X > n X d n = n d d The
Hard-Margin Support Vector Machines
Hard-Margin Support Vector Machines aaacaxicbzdlssnafiyn9vbjlepk3ay2gicupasvu4iblxuaw2hjmuwn7ddjjmxm1bkcg1/fjqsvt76fo9/gazqfvn8y+pjpozw5vx8zkpvtfxmlhcwl5zxyqrm2vrg5zw3vxmsoezi4ogkr6phieky5crvvjhriqvdom9l2xxftevuwcekj3lktmhghgniauiyutvrwxtvme34a77kbvg73gtygpjsrfati1+xc8c84bvraowbf+uwnipyehcvmkjrdx46vlykhkgykm3ujjdhcyzqkxy0chur6ax5cbg+1m4bbjptjcubuz4kuhvjoql93hkin5hxtav5x6yyqopnsyuneey5ni4keqrxbar5wqaxbik00icyo/iveiyqqvjo1u4fgzj/8f9x67bzmxnurjzmijtlybwfgcdjgfdtajwgcf2dwaj7ac3g1ho1n4814n7wwjgjmf/ys8fenfycuzq==
Roland HINNION. Introduction
REPORTS ON MATHEMATICAL LOGIC 47 (2012), 115 124 DOI:10.4467/20842589RM.12.005.0686 Roland HINNION ULTRAFILTERS (WITH DENSE ELEMENTS) OVER CLOSURE SPACES A b s t r a c t. Several notions and results that
SSW1.1, HFW Fry #20, Zeno #25 Benchmark: Qtr.1. Fry #65, Zeno #67. like
SSW1.1, HFW Fry #20, Zeno #25 Benchmark: Qtr.1 I SSW1.1, HFW Fry #65, Zeno #67 Benchmark: Qtr.1 like SSW1.2, HFW Fry #47, Zeno #59 Benchmark: Qtr.1 do SSW1.2, HFW Fry #5, Zeno #4 Benchmark: Qtr.1 to SSW1.2,
Stargard Szczecinski i okolice (Polish Edition)
Stargard Szczecinski i okolice (Polish Edition) Janusz Leszek Jurkiewicz Click here if your download doesn"t start automatically Stargard Szczecinski i okolice (Polish Edition) Janusz Leszek Jurkiewicz
www.irs.gov/form990. If "Yes," complete Schedule A Schedule B, Schedule of Contributors If "Yes," complete Schedule C, Part I If "Yes," complete Schedule C, Part II If "Yes," complete Schedule C, Part
EXAMPLES OF CABRI GEOMETRE II APPLICATION IN GEOMETRIC SCIENTIFIC RESEARCH
Anna BŁACH Centre of Geometry and Engineering Graphics Silesian University of Technology in Gliwice EXAMPLES OF CABRI GEOMETRE II APPLICATION IN GEOMETRIC SCIENTIFIC RESEARCH Introduction Computer techniques
Karpacz, plan miasta 1:10 000: Panorama Karkonoszy, mapa szlakow turystycznych (Polish Edition)
Karpacz, plan miasta 1:10 000: Panorama Karkonoszy, mapa szlakow turystycznych (Polish Edition) J Krupski Click here if your download doesn"t start automatically Karpacz, plan miasta 1:10 000: Panorama
www.irs.gov/form990. If "Yes," complete Schedule A Schedule B, Schedule of Contributors If "Yes," complete Schedule C, Part I If "Yes," complete Schedule C, Part II If "Yes," complete Schedule C, Part
Rachunek lambda, zima
Rachunek lambda, zima 2015-16 Wykład 2 12 października 2015 Tydzień temu: Własność Churcha-Rossera (CR) Jeśli a b i a c, to istnieje takie d, że b d i c d. Tydzień temu: Własność Churcha-Rossera (CR) Jeśli
Miedzy legenda a historia: Szlakiem piastowskim z Poznania do Gniezna (Biblioteka Kroniki Wielkopolski) (Polish Edition)
Miedzy legenda a historia: Szlakiem piastowskim z Poznania do Gniezna (Biblioteka Kroniki Wielkopolski) (Polish Edition) Piotr Maluskiewicz Click here if your download doesn"t start automatically Miedzy
Stability of Tikhonov Regularization Class 07, March 2003 Alex Rakhlin
Stability of Tikhonov Regularization 9.520 Class 07, March 2003 Alex Rakhlin Plan Review of Stability Bounds Stability of Tikhonov Regularization Algorithms Uniform Stability Review notation: S = {z 1,...,
O przecinkach i nie tylko
O przecinkach i nie tylko Jerzy Trzeciak Dział Wydawnictw IMPAN publ@impan.pl https://www.impan.pl/pl/wydawnictwa/dla-autorow 7 sierpnia 2018 Przecinek Sformułuję najpierw kilka zasad, którymi warto się
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition)
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition) Robert Respondowski Click here if your download doesn"t start automatically Wojewodztwo Koszalinskie:
MaPlan Sp. z O.O. Click here if your download doesn"t start automatically
Mierzeja Wislana, mapa turystyczna 1:50 000: Mikoszewo, Jantar, Stegna, Sztutowo, Katy Rybackie, Przebrno, Krynica Morska, Piaski, Frombork =... = Carte touristique (Polish Edition) MaPlan Sp. z O.O Click
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition)
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition) Robert Respondowski Click here if your download doesn"t start automatically Wojewodztwo Koszalinskie:
Rozpoznawanie twarzy metodą PCA Michał Bereta 1. Testowanie statystycznej istotności różnic między jakością klasyfikatorów
Rozpoznawanie twarzy metodą PCA Michał Bereta www.michalbereta.pl 1. Testowanie statystycznej istotności różnic między jakością klasyfikatorów Wiemy, że możemy porównywad klasyfikatory np. za pomocą kroswalidacji.
ARNOLD. EDUKACJA KULTURYSTY (POLSKA WERSJA JEZYKOWA) BY DOUGLAS KENT HALL
Read Online and Download Ebook ARNOLD. EDUKACJA KULTURYSTY (POLSKA WERSJA JEZYKOWA) BY DOUGLAS KENT HALL DOWNLOAD EBOOK : ARNOLD. EDUKACJA KULTURYSTY (POLSKA WERSJA Click link bellow and free register
EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION
Glasgow Math. J.: page 1 of 12. C Glasgow Mathematical Journal Trust 2012. doi:10.1017/s0017089512000353. EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION SEAN SATHER-WAGSTAFF Mathematics Department,
Steeple #3: Gödel s Silver Blaze Theorem. Selmer Bringsjord Are Humans Rational? Dec RPI Troy NY USA
Steeple #3: Gödel s Silver Blaze Theorem Selmer Bringsjord Are Humans Rational? Dec 6 2018 RPI Troy NY USA Gödels Great Theorems (OUP) by Selmer Bringsjord Introduction ( The Wager ) Brief Preliminaries
Linear Classification and Logistic Regression. Pascal Fua IC-CVLab
Linear Classification and Logistic Regression Pascal Fua IC-CVLab 1 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
Katowice, plan miasta: Skala 1: = City map = Stadtplan (Polish Edition)
Katowice, plan miasta: Skala 1:20 000 = City map = Stadtplan (Polish Edition) Polskie Przedsiebiorstwo Wydawnictw Kartograficznych im. Eugeniusza Romera Click here if your download doesn"t start automatically
TTIC 31210: Advanced Natural Language Processing. Kevin Gimpel Spring Lecture 8: Structured PredicCon 2
TTIC 31210: Advanced Natural Language Processing Kevin Gimpel Spring 2019 Lecture 8: Structured PredicCon 2 1 Roadmap intro (1 lecture) deep learning for NLP (5 lectures) structured predic+on (4 lectures)
Karpacz, plan miasta 1:10 000: Panorama Karkonoszy, mapa szlakow turystycznych (Polish Edition)
Karpacz, plan miasta 1:10 000: Panorama Karkonoszy, mapa szlakow turystycznych (Polish Edition) J Krupski Click here if your download doesn"t start automatically Karpacz, plan miasta 1:10 000: Panorama
Installation of EuroCert software for qualified electronic signature
Installation of EuroCert software for qualified electronic signature for Microsoft Windows systems Warsaw 28.08.2019 Content 1. Downloading and running the software for the e-signature... 3 a) Installer
www.irs.gov/form990. If "Yes," complete Schedule A Schedule B, Schedule of Contributors If "Yes," complete Schedule C, Part I If "Yes," complete Schedule C, Part II If "Yes," complete Schedule C, Part
Proposal of thesis topic for mgr in. (MSE) programme in Telecommunications and Computer Science
Proposal of thesis topic for mgr in (MSE) programme 1 Topic: Monte Carlo Method used for a prognosis of a selected technological process 2 Supervisor: Dr in Małgorzata Langer 3 Auxiliary supervisor: 4
Analysis of Movie Profitability STAT 469 IN CLASS ANALYSIS #2
Analysis of Movie Profitability STAT 469 IN CLASS ANALYSIS #2 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
Few-fermion thermometry
Few-fermion thermometry Phys. Rev. A 97, 063619 (2018) Tomasz Sowiński Institute of Physics of the Polish Academy of Sciences Co-authors: Marcin Płodzień Rafał Demkowicz-Dobrzański FEW-BODY PROBLEMS FewBody.ifpan.edu.pl
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition)
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition) Robert Respondowski Click here if your download doesn"t start automatically Wojewodztwo Koszalinskie:
ERASMUS + : Trail of extinct and active volcanoes, earthquakes through Europe. SURVEY TO STUDENTS.
ERASMUS + : Trail of extinct and active volcanoes, earthquakes through Europe. SURVEY TO STUDENTS. Strona 1 1. Please give one answer. I am: Students involved in project 69% 18 Student not involved in
!850016! www.irs.gov/form8879eo. e-file www.irs.gov/form990. If "Yes," complete Schedule A Schedule B, Schedule of Contributors If "Yes," complete Schedule C, Part I If "Yes," complete Schedule C,
SubVersion. Piotr Mikulski. SubVersion. P. Mikulski. Co to jest subversion? Zalety SubVersion. Wady SubVersion. Inne różnice SubVersion i CVS
Piotr Mikulski 2006 Subversion is a free/open-source version control system. That is, Subversion manages files and directories over time. A tree of files is placed into a central repository. The repository
Permutability Class of a Semigroup
Journal of Algebra 226, 295 310 (2000) doi:10.1006/jabr.1999.8174, available online at http://www.idealibrary.com on Permutability Class of a Semigroup Andrzej Kisielewicz 1 Mathematical Institute, University
OBWIESZCZENIE MINISTRA INFRASTRUKTURY. z dnia 18 kwietnia 2005 r.
OBWIESZCZENIE MINISTRA INFRASTRUKTURY z dnia 18 kwietnia 2005 r. w sprawie wejścia w życie umowy wielostronnej M 163 zawartej na podstawie Umowy europejskiej dotyczącej międzynarodowego przewozu drogowego
Blow-Up: Photographs in the Time of Tumult; Black and White Photography Festival Zakopane Warszawa 2002 / Powiekszenie: Fotografie w czasach zgielku
Blow-Up: Photographs in the Time of Tumult; Black and White Photography Festival Zakopane Warszawa 2002 / Powiekszenie: Fotografie w czasach zgielku Juliusz and Maciej Zalewski eds. and A. D. Coleman et
DODATKOWE ĆWICZENIA EGZAMINACYJNE
I.1. X Have a nice day! Y a) Good idea b) See you soon c) The same to you I.2. X: This is my new computer. Y: Wow! Can I have a look at the Internet? X: a) Thank you b) Go ahead c) Let me try I.3. X: What
www.irs.gov/form990. If "Yes," complete Schedule A Schedule B, Schedule of Contributors If "Yes," complete Schedule C, Part I If "Yes," complete Schedule C, Part II If "Yes," complete Schedule C, Part
18. Przydatne zwroty podczas egzaminu ustnego. 19. Mo liwe pytania egzaminatora i przyk³adowe odpowiedzi egzaminowanego
18. Przydatne zwroty podczas egzaminu ustnego I m sorry, could you repeat that, please? - Przepraszam, czy mo na prosiæ o powtórzenie? I m sorry, I don t understand. - Przepraszam, nie rozumiem. Did you
EGZAMIN MATURALNY Z JĘZYKA ANGIELSKIEGO POZIOM ROZSZERZONY MAJ 2010 CZĘŚĆ I. Czas pracy: 120 minut. Liczba punktów do uzyskania: 23 WPISUJE ZDAJĄCY
Centralna Komisja Egzaminacyjna Arkusz zawiera informacje prawnie chronione do momentu rozpoczęcia egzaminu. Układ graficzny CKE 2010 KOD WPISUJE ZDAJĄCY PESEL Miejsce na naklejkę z kodem dysleksja EGZAMIN
OpenPoland.net API Documentation
OpenPoland.net API Documentation Release 1.0 Michał Gryczka July 11, 2014 Contents 1 REST API tokens: 3 1.1 How to get a token............................................ 3 2 REST API : search for assets
Patients price acceptance SELECTED FINDINGS
Patients price acceptance SELECTED FINDINGS October 2015 Summary With growing economy and Poles benefiting from this growth, perception of prices changes - this is also true for pharmaceuticals It may
y = The Chain Rule Show all work. No calculator unless otherwise stated. If asked to Explain your answer, write in complete sentences.
The Chain Rule Show all work. No calculator unless otherwise stated. If asked to Eplain your answer, write in complete sentences. 1. Find the derivative of the functions y 7 (b) (a) ( ) y t 1 + t 1 (c)
Karpacz, plan miasta 1:10 000: Panorama Karkonoszy, mapa szlakow turystycznych (Polish Edition)
Karpacz, plan miasta 1:10 000: Panorama Karkonoszy, mapa szlakow turystycznych (Polish Edition) J Krupski Click here if your download doesn"t start automatically Karpacz, plan miasta 1:10 000: Panorama
deep learning for NLP (5 lectures)
TTIC 31210: Advanced Natural Language Processing Kevin Gimpel Spring 2019 Lecture 6: Finish Transformers; Sequence- to- Sequence Modeling and AJenKon 1 Roadmap intro (1 lecture) deep learning for NLP (5
EPS. Erasmus Policy Statement
Wyższa Szkoła Biznesu i Przedsiębiorczości Ostrowiec Świętokrzyski College of Business and Entrepreneurship EPS Erasmus Policy Statement Deklaracja Polityki Erasmusa 2014-2020 EN The institution is located
Wybrzeze Baltyku, mapa turystyczna 1: (Polish Edition)
Wybrzeze Baltyku, mapa turystyczna 1:50 000 (Polish Edition) Click here if your download doesn"t start automatically Wybrzeze Baltyku, mapa turystyczna 1:50 000 (Polish Edition) Wybrzeze Baltyku, mapa
www.irs.gov/form990. If "Yes," complete Schedule A Schedule B, Schedule of Contributors If "Yes," complete Schedule C, Part I If "Yes," complete Schedule C, Part II If "Yes," complete Schedule C, Part
General Certificate of Education Ordinary Level ADDITIONAL MATHEMATICS 4037/12
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level www.xtremepapers.com *6378719168* ADDITIONAL MATHEMATICS 4037/12 Paper 1 May/June 2013 2 hours Candidates
Title: On the curl of singular completely continous vector fields in Banach spaces
Title: On the curl of singular completely continous vector fields in Banach spaces Author: Adam Bielecki, Tadeusz Dłotko Citation style: Bielecki Adam, Dłotko Tadeusz. (1973). On the curl of singular completely
Miedzy legenda a historia: Szlakiem piastowskim z Poznania do Gniezna (Biblioteka Kroniki Wielkopolski) (Polish Edition)
Miedzy legenda a historia: Szlakiem piastowskim z Poznania do Gniezna (Biblioteka Kroniki Wielkopolski) (Polish Edition) Piotr Maluskiewicz Click here if your download doesn"t start automatically Miedzy
Jak zasada Pareto może pomóc Ci w nauce języków obcych?
Jak zasada Pareto może pomóc Ci w nauce języków obcych? Artykuł pobrano ze strony eioba.pl Pokazuje, jak zastosowanie zasady Pareto może usprawnić Twoją naukę angielskiego. Słynna zasada Pareto mówi o
Emilka szuka swojej gwiazdy / Emily Climbs (Emily, #2)
Emilka szuka swojej gwiazdy / Emily Climbs (Emily, #2) Click here if your download doesn"t start automatically Emilka szuka swojej gwiazdy / Emily Climbs (Emily, #2) Emilka szuka swojej gwiazdy / Emily
Surname. Other Names. For Examiner s Use Centre Number. Candidate Number. Candidate Signature
A Surname _ Other Names For Examiner s Use Centre Number Candidate Number Candidate Signature Polish Unit 1 PLSH1 General Certificate of Education Advanced Subsidiary Examination June 2014 Reading and
Poland) Wydawnictwo "Gea" (Warsaw. Click here if your download doesn"t start automatically
Suwalski Park Krajobrazowy i okolice 1:50 000, mapa turystyczno-krajoznawcza =: Suwalki Landscape Park, tourist map = Suwalki Naturpark,... narodowe i krajobrazowe) (Polish Edition) Click here if your
Dolny Slask 1: , mapa turystycznosamochodowa: Plan Wroclawia (Polish Edition)
Dolny Slask 1:300 000, mapa turystycznosamochodowa: Plan Wroclawia (Polish Edition) Click here if your download doesn"t start automatically Dolny Slask 1:300 000, mapa turystyczno-samochodowa: Plan Wroclawia
Extraclass. Football Men. Season 2009/10 - Autumn round
Extraclass Football Men Season 2009/10 - Autumn round Invitation Dear All, On the date of 29th July starts the new season of Polish Extraclass. There will be live coverage form all the matches on Canal+
Twierdzenie Hilberta o nieujemnie określonych formach ternarnych stopnia 4
Twierdzenie Hilberta o nieujemnie określonych formach ternarnych stopnia 4 Strona 1 z 23 Andrzej Sładek, Instytut Matematyki UŚl sladek@math.us.edu.pl Letnia Szkoła Instytutu Matematyki 20-23 września
Pielgrzymka do Ojczyzny: Przemowienia i homilie Ojca Swietego Jana Pawla II (Jan Pawel II-- pierwszy Polak na Stolicy Piotrowej) (Polish Edition)
Pielgrzymka do Ojczyzny: Przemowienia i homilie Ojca Swietego Jana Pawla II (Jan Pawel II-- pierwszy Polak na Stolicy Piotrowej) (Polish Edition) Click here if your download doesn"t start automatically
Fig 5 Spectrograms of the original signal (top) extracted shaft-related GAD components (middle) and
Fig 4 Measured vibration signal (top). Blue original signal. Red component related to periodic excitation of resonances and noise. Green component related. Rotational speed profile used for experiment
Leba, Rowy, Ustka, Slowinski Park Narodowy, plany miast, mapa turystyczna =: Tourist map = Touristenkarte (Polish Edition)
Leba, Rowy, Ustka, Slowinski Park Narodowy, plany miast, mapa turystyczna =: Tourist map = Touristenkarte (Polish Edition) FotKart s.c Click here if your download doesn"t start automatically Leba, Rowy,
Sargent Opens Sonairte Farmers' Market
Sargent Opens Sonairte Farmers' Market 31 March, 2008 1V8VIZSV7EVKIRX8(1MRMWXIVSJ7XEXIEXXLI(ITEVXQIRXSJ%KVMGYPXYVI *MWLIVMIWERH*SSHTIVJSVQIHXLISJJMGMEPSTIRMRKSJXLI7SREMVXI*EVQIVW 1EVOIXMR0E]XS[R'S1IEXL
Wydział Fizyki, Astronomii i Informatyki Stosowanej Uniwersytet Mikołaja Kopernika w Toruniu
IONS-14 / OPTO Meeting For Young Researchers 2013 Khet Tournament On 3-6 July 2013 at the Faculty of Physics, Astronomy and Informatics of Nicolaus Copernicus University in Torun (Poland) there were two
European Crime Prevention Award (ECPA) Annex I - new version 2014
European Crime Prevention Award (ECPA) Annex I - new version 2014 Załącznik nr 1 General information (Informacje ogólne) 1. Please specify your country. (Kraj pochodzenia:) 2. Is this your country s ECPA
Polska Szkoła Weekendowa, Arklow, Co. Wicklow KWESTIONRIUSZ OSOBOWY DZIECKA CHILD RECORD FORM
KWESTIONRIUSZ OSOBOWY DZIECKA CHILD RECORD FORM 1. Imię i nazwisko dziecka / Child's name... 2. Adres / Address... 3. Data urodzenia / Date of birth... 4. Imię i nazwisko matki /Mother's name... 5. Adres
Realizacja systemów wbudowanych (embeded systems) w strukturach PSoC (Programmable System on Chip)
Realizacja systemów wbudowanych (embeded systems) w strukturach PSoC (Programmable System on Chip) Embeded systems Architektura układów PSoC (Cypress) Możliwości bloków cyfrowych i analogowych Narzędzia
Ankiety Nowe funkcje! Pomoc magda.szewczyk@slo-wroc.pl. magda.szewczyk@slo-wroc.pl. Twoje konto Wyloguj. BIODIVERSITY OF RIVERS: Survey to students
Ankiety Nowe funkcje! Pomoc magda.szewczyk@slo-wroc.pl Back Twoje konto Wyloguj magda.szewczyk@slo-wroc.pl BIODIVERSITY OF RIVERS: Survey to students Tworzenie ankiety Udostępnianie Analiza (55) Wyniki
Previously on CSCI 4622
More Naïve Bayes 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
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition)
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition) Robert Respondowski Click here if your download doesn"t start automatically Wojewodztwo Koszalinskie:
Machine Learning for Data Science (CS4786) Lecture 11. Spectral Embedding + Clustering
Machine Learning for Data Science (CS4786) Lecture 11 Spectral Embedding + Clustering MOTIVATING EXAMPLE What can you say from this network? MOTIVATING EXAMPLE How about now? THOUGHT EXPERIMENT For each
ZDANIA ANGIELSKIE W PARAFRAZIE
MACIEJ MATASEK ZDANIA ANGIELSKIE W PARAFRAZIE HANDYBOOKS 1 Copyright by Wydawnictwo HANDYBOOKS Poznań 2014 Wszelkie prawa zastrzeżone. Każda reprodukcja lub adaptacja całości bądź części niniejszej publikacji,
Standardized Test Practice
Standardized Test Practice 1. Which of the following is the length of a three-dimensional diagonal of the figure shown? a. 4.69 units b. 13.27 units c. 13.93 units 3 d. 16.25 units 8 2. Which of the following
CS 6170: Computational Topology, Spring 2019 Lecture 09
CS 6170: Computtionl Topology, Spring 2019 Lecture 09 Topologicl Dt Anlysis for Dt Scientists Dr. Bei Wng School of Computing Scientific Computing nd Imging Institute (SCI) University of Uth www.sci.uth.edu/~beiwng
EGZAMIN MATURALNY Z JĘZYKA ANGIELSKIEGO POZIOM ROZSZERZONY MAJ 2010 CZĘŚĆ I. Czas pracy: 120 minut. Liczba punktów do uzyskania: 23 WPISUJE ZDAJĄCY
Centralna Komisja Egzaminacyjna Arkusz zawiera informacje prawnie chronione do momentu rozpoczęcia egzaminu. Układ graficzny CKE 2010 KOD WPISUJE ZDAJĄCY PESEL Miejsce na naklejkę z kodem dysleksja EGZAMIN
DUAL SIMILARITY OF VOLTAGE TO CURRENT AND CURRENT TO VOLTAGE TRANSFER FUNCTION OF HYBRID ACTIVE TWO- PORTS WITH CONVERSION
ELEKTRYKA 0 Zeszyt (9) Rok LX Andrzej KUKIEŁKA Politechnika Śląska w Gliwicach DUAL SIMILARITY OF VOLTAGE TO CURRENT AND CURRENT TO VOLTAGE TRANSFER FUNCTION OF HYBRID ACTIVE TWO- PORTS WITH CONVERSION
DOI: / /32/37
. 2015. 4 (32) 1:18 DOI: 10.17223/1998863 /32/37 -,,. - -. :,,,,., -, -.,.-.,.,.,. -., -,.,,., -, 70 80. (.,.,. ),, -,.,, -,, (1886 1980).,.,, (.,.,..), -, -,,,, ; -, - 346, -,.. :, -, -,,,,,.,,, -,,,
Wroclaw, plan nowy: Nowe ulice, 1:22500, sygnalizacja swietlna, wysokosc wiaduktow : Debica = City plan (Polish Edition)
Wroclaw, plan nowy: Nowe ulice, 1:22500, sygnalizacja swietlna, wysokosc wiaduktow : Debica = City plan (Polish Edition) Wydawnictwo "Demart" s.c Click here if your download doesn"t start automatically
SNP SNP Business Partner Data Checker. Prezentacja produktu
SNP SNP Business Partner Data Checker Prezentacja produktu Istota rozwiązania SNP SNP Business Partner Data Checker Celem produktu SNP SNP Business Partner Data Checker jest umożliwienie sprawdzania nazwy
www.irs.gov/form8879eo. e-file www.irs.gov/form990. If "Yes," complete Schedule A Schedule B, Schedule of Contributors If "Yes," complete Schedule C, Part I If "Yes," complete Schedule C, Part II If "Yes,"
Probability definition
Probability Probability definition Probability of an event P(A) denotes the frequency of it s occurrence in a long (infinite) series of repeats P( A) = A Ω Ω A 0 P It s s represented by a number in the
PSB dla masazystow. Praca Zbiorowa. Click here if your download doesn"t start automatically
PSB dla masazystow Praca Zbiorowa Click here if your download doesn"t start automatically PSB dla masazystow Praca Zbiorowa PSB dla masazystow Praca Zbiorowa Podrecznik wydany w formie kieszonkowego przewodnika,
PLSH1 (JUN14PLSH101) General Certificate of Education Advanced Subsidiary Examination June 2014. Reading and Writing TOTAL
Centre Number Surname Candidate Number For Examiner s Use Other Names Candidate Signature Examiner s Initials Section Mark Polish Unit 1 Reading and Writing General Certificate of Education Advanced Subsidiary
Machine Learning for Data Science (CS4786) Lecture 24. Differential Privacy and Re-useable Holdout
Machine Learning for Data Science (CS4786) Lecture 24 Differential Privacy and Re-useable Holdout Defining Privacy Defining Privacy Dataset + Defining Privacy Dataset + Learning Algorithm Distribution
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition)
Wojewodztwo Koszalinskie: Obiekty i walory krajoznawcze (Inwentaryzacja krajoznawcza Polski) (Polish Edition) Robert Respondowski Click here if your download doesn"t start automatically Wojewodztwo Koszalinskie:
Immigration Studying. Studying - University. Stating that you want to enroll. Stating that you want to apply for a course.
- University I would like to enroll at a university. Stating that you want to enroll I want to apply for course. Stating that you want to apply for a course an undergraduate a postgraduate a PhD a full-time
Formularz recenzji magazynu. Journal of Corporate Responsibility and Leadership Review Form
Formularz recenzji magazynu Review Form Identyfikator magazynu/ Journal identification number: Tytuł artykułu/ Paper title: Recenzent/ Reviewer: (imię i nazwisko, stopień naukowy/name and surname, academic
Nazwa projektu: Kreatywni i innowacyjni uczniowie konkurencyjni na rynku pracy
Nazwa projektu: Kreatywni i innowacyjni uczniowie konkurencyjni na rynku pracy DZIAŁANIE 3.2 EDUKACJA OGÓLNA PODDZIAŁANIE 3.2.1 JAKOŚĆ EDUKACJI OGÓLNEJ Projekt współfinansowany przez Unię Europejską w
Domy inaczej pomyślane A different type of housing CEZARY SANKOWSKI
Domy inaczej pomyślane A different type of housing CEZARY SANKOWSKI O tym, dlaczego warto budować pasywnie, komu budownictwo pasywne się opłaca, a kto się go boi, z architektem, Cezarym Sankowskim, rozmawia
Wroclaw, plan nowy: Nowe ulice, 1:22500, sygnalizacja swietlna, wysokosc wiaduktow : Debica = City plan (Polish Edition)
Wroclaw, plan nowy: Nowe ulice, 1:22500, sygnalizacja swietlna, wysokosc wiaduktow : Debica = City plan (Polish Edition) Wydawnictwo "Demart" s.c Click here if your download doesn"t start automatically
17-18 września 2016 Spółka Limited w UK. Jako Wehikuł Inwestycyjny. Marek Niedźwiedź. InvestCamp 2016 PL
17-18 września 2016 Spółka Limited w UK Jako Wehikuł Inwestycyjny InvestCamp 2016 PL Marek Niedźwiedź A G E N D A Dlaczego Spółka Ltd? Stabilność Bezpieczeństwo Narzędzia 1. Stabilność brytyjskiego systemu
Zmiany techniczne wprowadzone w wersji Comarch ERP Altum
Zmiany techniczne wprowadzone w wersji 2018.2 Copyright 2016 COMARCH SA Wszelkie prawa zastrzeżone Nieautoryzowane rozpowszechnianie całości lub fragmentu niniejszej publikacji w jakiejkolwiek postaci
Polski Krok Po Kroku: Tablice Gramatyczne (Polish Edition) By Anna Stelmach
Polski Krok Po Kroku: Tablice Gramatyczne (Polish Edition) By Anna Stelmach If you are looking for the ebook by Anna Stelmach Polski krok po kroku: Tablice gramatyczne (Polish Edition) in pdf form, in
ZGŁOSZENIE WSPÓLNEGO POLSKO -. PROJEKTU NA LATA: APPLICATION FOR A JOINT POLISH -... PROJECT FOR THE YEARS:.
ZGŁOSZENIE WSPÓLNEGO POLSKO -. PROJEKTU NA LATA: APPLICATION FOR A JOINT POLISH -... PROJECT FOR THE YEARS:. W RAMACH POROZUMIENIA O WSPÓŁPRACY NAUKOWEJ MIĘDZY POLSKĄ AKADEMIĄ NAUK I... UNDER THE AGREEMENT