Pathologies in cohomology of non-paracompact Hausdorff spaces
aa r X i v : . [ m a t h . A T ] S e p PATHOLOGIES IN COHOMOLOGY OF NON-PARACOMPACTHAUSDORFF SPACES
STEFAN SCHR ¨OER
Revised version, 21 June 2013
Abstract.
We construct a non-paracompact Hausdorff space for which ˇCechcohomology does not coincide with sheaf cohomology. Moreover, the sheafof continuous real-valued functions is neither soft nor acyclic, and our spaceadmits non-numerable principal bundles.
Introduction
Recall that a topological space X is called paracompact if it is Hausdorff, and eachopen covering admits a refinement that is locally finite. This notion was introducedby Dieudonn´e [4] as early as 1944 and has turned out to be extremely useful ingeneral topology and sheaf theory. For example, Godement showed that ˇCechcohomology coincides with sheaf cohomology on paracompact spaces ([6], Theorem5.10.1). For general spaces, all that can be said is that there is a spectral sequenceˇ H p ( X, H q ( F )) = ⇒ H p + q ( X, F )computing the “true” sheaf cohomology from the ˇCech cohomology of the presheavesof sheaf cohomology (loc. cit., Theorem 5.9.1). Grothendieck observed that formany irreducible spaces, for example X = C with the Zariski topology, this spec-tral sequence does not degenerate for suitable F , such that ˇCech cohomology doesnot coincide with sheaf cohomology ([7], page 178). On the other hand, Artin [1]established that for “most” separated schemes, ˇCech cohomology agrees with sheafcohomology when computed in the ´etale topology.Although the known counterexamples are very common in the realm of algebraicgeometry, they are perhaps not so natural from the standpoint of algebraic orgeneral topology, since the spaces are not Hausdorff. In my opinion, it would bedesirable to have further counterexamples satisfying the Hausdorff axiom, the moreso in light of Artin’s result.The goal of this note is to provide such a space. The construction roughly goesas follows: We start with an infinite wedge sum X = W ∞ i =1 D of closed 2-disks,and replace the CW-topology at the intersection of the 2-disks by some coarsertopology. This topology is choose fine enough to keep the space Hausdorff, yetcoarse enough so that a variant of Grothendieck’s argument holds true.It turns out that our space has other pathological features as well: The sheafof continuous real-valued functions is neither soft nor acyclic. Although the spaceis contractible, it carries nontrivial principal S -bundles. These are necessarilynon-numerable, whence do not come from the universal bundle. Mathematics Subject Classification. The construction
We start by constructing an infinite 2-dimensional CW-complex X . Its 0-skeletonis a sequence e n , n ≥ x = e will play a specialrole throughout, and we shall call it the origin . To form the 1-skeleton X , weconnect the origin to each e n , n ≥ e ± n . To complete theconstruction, we choose homeomorphisms ϕ n : S −→ e ∪ e n ∪ e n ∪ e − n ⊂ X , and use these as attaching maps for the 2-cells e n , n ≥
1. This gives an infinite2-dimensional CW-complex X , which one may visualize as follows. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) the origin x other 0−cells Figure 1: The CW-complex X Being a CW-complex, the space X is paracompact [10]. With our goal in mindwe now replace the CW-topology by some coarser topology: Let τ be the collectionof all subsets U ⊂ X that are open in the CW-topology, and either do not containthe origin x , or contain almost all subsets e n r e n , which are closed 2-cells with a 0-cell removed. This collection of subsets obviously satisfies the axioms of a topology,and we call this topology τ the coarser topology . Here and throughout, almost allmeans all but finitely many. The set X , endowed with the coarser topology, isdenoted X crs . Proposition 1.1.
The space X crs is Hausdorff but not paracompact.Proof. Clearly, the identity map X → X crs is continuous, and becomes a homeo-morphism outside the origin. Thus X crs is Hausdorff outside the origin x . Given y = x , we choose two disjoint open neighborhoods x ∈ U , y ∈ V on the CW-complex X . By shrinking V , we may assume that V intersects only one closed2-cell. By enlarging U , we may assume that U contains all remaining closed 2-cells,while staying disjoint from V . Then U, V are open in the coarse topology, thus X crs is Hausdorff.To see that the space is not paracompact, let U ⊂ X crs be the complement of S n ≥ e n , and U n ⊂ X by the open subset e n r { x } . This gives an open covering X crs = S n ≥ U n . Every refinement of this covering fails to be locally finite: For each n ≥
1, let U ′ n be a member of such a refinement that contains e n . Clearly, U ′ n ⊂ U n ,whence the U ′ n are pairwise different. By definition of the coarser topology, eachneighborhood of the origin contains almost all e n r e n , therefore intersects almostall U ′ n . Hence our space is not paracompact. (cid:3) ATHOLOGIES IN COHOMOLOGY 3
Remark 1.2.
The space X crs is not regular : Consider the origin x and the closedset A = S n ≥ e n . Then every open neighborhood of x intersects every open neigh-borhood of A . On the other hand, the space X crs is pointwise paracompact , a prop-erty also called metacompactness : Every open covering admits a refinement that ispointwise finite. Clearly, each closed 2-cell e n ⊂ X crs is compact, hence X crs is acountable union of compacta, in other words, our space is σ -compact . In particular,it is Lindel¨of , which means that every open covering has a countable subcovering.The reader may consult Steen and Seebach [11] for other counterexamples in thisdirection.
Remark 1.3.
The
Kelley topology (also called the compactly generated topology )on a space Y consists of those subsets V ⊂ Y such that V ∩ K ⊂ K is open for eachcompact subset K ⊂ Y . This topology plays a role for infinite CW-complexes, forexample, to define products. One easily checks that each compact subset K ⊂ X crs is also compact with respect to the CW-topology. From this it follows that theKelley topology of X crs coincides with the CW-topology.2. ˇCech and sheaf cohomology Let F be an abelian sheaf on a topological space Y . Then one has sheaf co-homology groups H p ( Y, F ), which are defined via global sections of injective res-olutions, and ˇCech cohomology groups ˇ H p ( Y, F ), which are computed in terms ofopen coverings and local sections. The two types of cohomology groups are re-lated by a spectral sequence ˇ H p ( Y, H q ( F )) ⇒ H p + q ( Y, F ). For details, we referto Grothendieck’s exposition [7] and Godement’s monograph [6]. A basic fact insheaf theory states that the canonical map ˇ H ( Y, F ) → H ( Y, F ) is bijective, andwe have a short exact sequence0 −→ ˇ H ( Y, F ) −→ H ( Y, F ) −→ ˇ H ( Y, H ( F )) −→ , compare [7], page 177. Thus ˇCech cohomology does not coincide with sheaf coho-mology provided ˇ H ( Y, H ( F )) = 0.Our task is therefore to find such a situation. Consider the CW-complex X and the space X crs constructed in the preceding section. The following fact will beuseful: Lemma 2.1.
For each open subset V ⊂ X crs , the sheaf cohomology groups H p ( V, Z ) , p ≥ are the same, whether computed in the CW-topology or in the coarser topol-ogy.Proof. Let i : X → X crs be the identity map, which is continuous. We have acanonical map Z X crs → i ∗ ( Z X ) of abelian sheaves, where the left hand side is thesheaf of locally constant integer-valued functions on X crs , and the right hand sideis the direct image sheaf of the corresponding sheaf on X . We first check that thismap is bijective. The question is local on X crs , and bijectivity is obvious outsidethe origin. Injectivity holds because the mapping i is surjective. Since there arearbitrarily small open neighborhoods x ∈ V ⊂ X crs that are pathconnected andhence connected in the CW-topology, the canonical map Z X crs ,x → i ∗ ( Z X ) x isbijective as well.In light of the Leray–Serre spectral sequence H p ( X crs , R q i ∗ ( Z X )) = ⇒ H p + q ( X, Z X ) , STEFAN SCHR ¨OER it suffices to check that R p i ∗ ( Z X ) = 0 for all p >
0. This is again local, andholds for trivial reasons outside the origin. Since there are arbitrarily small openneighborhoods x ∈ V ⊂ X crs that are contractible in the CW-topology, and singularcohomology coincides with sheaf cohomology for CW-complexes ([2], Chapter III,Section 1), vanishing holds at the origin as well. (cid:3) Let U ⊂ X be the complement of the 1-skeleton X ⊂ X , that is, the union of all2-cells, and Z U be the abelian sheaf of locally constant integer-valued functions on U . Clearly, U is open in the coarser topology. Thus the inclusion map i : U → X crs is continuous. From this we obtain an abelian sheaf F = i ! ( Z U ) on X crs , called extension by zero . It is defined by the ruleΓ( V, F ) = ( Γ( V, Z U ) if V ⊂ U ;0 else , compare [8], Expose I. Its first cohomology is easily computed: Proposition 2.2.
Let V ⊂ X crs be an open subset with H ( V, Z ) = 0 . Then wehave a canonical identification H ( V, F ) = H ( V ∩ X , Z ) /H ( V, Z ) . Proof.
The short exact sequence 0 → F → Z X crs → Z X → H ( V, Z ) −→ H ( V ∩ X , Z ) −→ H ( V, F ) −→ H ( V, Z ) , and the result follows. (cid:3) For this sheaf, ˇCech cohomology does not coincide with sheaf cohomology, in arather drastic way:
Theorem 2.3.
For the abelian sheaf F = i ! ( Z U ) on the topological space X crs thegroup ˇ H ( X crs , H ( F )) is uncountable. In particular, the inclusion ˇ H ( X crs , F ) ⊂ H ( X crs , F ) is not bijective.Proof. Let U = ( U α ) α ∈ I be an open covering of X crs . By definition, the correspond-ing group ˇ H ( U , H ( F )) is the first cohomology of the complex(1) Y α H ( U α , F ) −→ Y α<β H ( U αβ , F ) −→ Y α<β<γ H ( U αβγ , F ) . Here we employ the usual abbreviation U αβ = U α ∩ U β et cetera. The coboundarymaps are the usual one, for example ( s α ) ( s β | U αβ − s α | U αβ ), and we have chosena total order on the index set I . By definition, ˇCech cohomology equalsˇ H ( X, H ( F )) = lim −→ U ˇ H ( U , H ( F )) , where the direct limit runs over all open coverings ordered by the refinement re-lation. For a precise definition of the maps in the direct system, and their well-definedness, we refer to [6], Chapter II, Section 5.7.In general, it can be difficult to control such direct limits. However, one mayrestrict to open coverings forming a cofinal subsystem. Therefore, we may assumethat our open covering satisfies the following five additional assumptions: (i) Each U α and the intersection U α ∩ X are, if nonempty, contractible in the CW-topology.(ii) Each 0-cell is contained in precisely one U α . (iii) If some U α contains a 0-cell e n , ATHOLOGIES IN COHOMOLOGY 5 n ≥
1, then it is contained in the corresponding closed 2-cell e n . (iv) We supposethat the index set I is well-ordered. This allows us to regard the natural numbers0 , , . . . ∈ I as indices. After reindexing, we stipulate that x ∈ U and e n ⊂ U n .(v) Finally, if a closed 2-cell e n is contained in U ∪ U n , then it is disjoint from allother U α .From now on, we only consider open coverings U satisfying these five condition.Choose m ≥ U contains all e n r e n , n ≥ m . Condition (i) implies thatfor V = U ∩ U n = U n r e n , n ≥ m we have H ( V, Z ) = 0, and furthermore H ( V ∩ X ) = Z ⊕ and H ( V, Z ) = Z , the latter sitting diagonally in the former. Note that this is the key step inGrothendieck’s argument [7], page 178. Now Proposition 2.2 gives us an identi-fication ∞ Y n = m H ( U ∩ U n ) = ∞ Y n = m Z . In light of Proposition 2.2, Condition (i) ensures that the term on the left in thecomplex (1) vanishes. Condition (ii) and (v) tell us that the triple intersections U ∩ U n ∩ U α are empty for n ≥ m and all indices α = 0 , n . The upshot is that wehave a canonical inclusion ∞ Y n = m Z = ∞ Y n = m H ( U ∩ U n , F ) ⊂ ˇ H ( U , H ( F )) . If U ′ is a refinement of U satisfying the same five conditions, the induced mapˇ H ( U , H ( F )) → ˇ H ( U ′ , H ( F )) restricts to the canonical projection ∞ Y n = m Z −→ ∞ Y n = m ′ Z on the subgroups considered above, where we tacitly choose m ′ ≥ m . Since formingdirect limits is exact, we obtain an inclusionlim −→ m ∞ Y n = m Z ⊂ lim −→ U ˇ H ( U , H ( F )) = ˇ H ( X, H ( F )) . Again using that forming direct limits is exact, we may rewrite the left hand sideas lim −→ m ∞ Y n =1 Z , m − Y n =1 Z ! = ∞ Y n =1 Z ! , lim −→ m m − Y n =1 Z ! = ∞ Y n =1 Z ! , ∞ M n =1 Z ! , which is uncountable. (cid:3) Continuous functions and principal bundles
We finally examine pathological properties of continuous functions and principalbundles on X crs . Let us write C X crs for the sheaf of continuous real-valued functionson X crs . Proposition 3.1.
We have H ( X crs , C X crs ) = 0 . STEFAN SCHR ¨OER
Proof.
Recall that ˇCech cohomology agrees with sheaf cohomology in degree one.Thus our task is to construct a nontrivial ˇCech cohomology class. Consider theopen covering U given by U = X crs r S n ≥ e n and U n = e n r { x } , n ≥
1. Choosegerms of continuous functions f n : ( U n , e n ) → R having an isolated zero at e n . Thenits reciprocal 1 /f n is defined on some open punctured neighborhood of e n ⊂ U n ,where it is necessarily unbounded. On the other hand, for any continuous function g : U → R there is some m ≥ g is bounded on S ∞ n = m e n ∩ U . Whence1 /f n cannot be written as the difference of continuous functions coming from U and U n , for n ≥ m . The same applies for any refinement U ′ satisfying the fiveconditions formulated in the proof for Theorem 2.3. The upshot is that for allrefinements U ′ with U ′ n sufficiently small, we obtain a well-defined tuple(1 /f n ) n ≥ m ∈ ∞ Y n = m H ( U ′ ∩ U ′ n , C X crs )that is a cocycle whose class in ˇ H ( U ′ , C X crs ) is nonzero. Recall that m ≥ U ′ contains e n r e n for all n ≥ m . Since this holds for all suchrefinements U ′ , it follows that the class in the direct limit ˇ H ( X crs , C X crs ) is nonzeroas well. (cid:3) Remark 3.2.
On normal spaces Y , the Uryson Lemma ensures that the sheaf C Y is soft , that is, the canonical map H ( Y, C Y ) → H ( A, i − ( C Y )) is surjective for allclosed subsets A , where i : A → Y denotes the inclusion map. According to [6],Chapter II, Theorem 4.4.3, soft sheaves on paracompact spaces Y are acyclic.For our space X crs , it is easy to check that the canonical map for the discreteclosed subset A = S n ≥ e n is not surjective. Summing up, the sheaf of continuousreal-valued functions on X crs is neither soft nor acyclic.Next consider the sheaf S X crs of continuous functions taking values in the circlegroup S = R / Z . It sits in the exponential sequence0 −→ Z −→ C X crs −→ S X crs −→ , where the map on the right is t e πit . From this we get an exact sequence H p ( X crs , Z ) −→ H p ( X crs , C X crs ) −→ H p ( X crs , S X crs ) −→ H p +1 ( X crs , Z ) . The outer terms vanish by Lemma 2.1, and we conclude that H p ( X crs , C X crs ) = H p ( X crs , S X crs )for all p >
0. From the preceding Proposition we get H ( X crs , S X crs ) = 0. In otherwords, there are nontrivial principal S -bundles over X crs . This is in stark contrastto the following fact: Proposition 3.3.
The space X crs is contractible.Proof. The CW-topology and the coarse topology induce the same topology on thecompact subsets e n ⊂ X crs , which are thus homeomorphic to the 2-disk. Choosehomotopies h n : e n × I → e n between the identity and the constant map to theorigin so that h n ( x, t ) = x for all t ∈ I , and h n ( y, t ) e n for all t > y . Thefirst condition ensures that the homotopies glue to a map h : X × I → X , which iscontinuous with respect to the CW-topology. From the second condition one easilyinfers that it remains continuous when regarded as a map h : X crs × I → X crs . Thus h is a homotopy from the identity on X crs to the constant map X crs → { x } . (cid:3) ATHOLOGIES IN COHOMOLOGY 7
Let G be a topological group. Recall that a G -principal bundle P → Y is called numerable if it can be trivialized on some numerable covering Y = S α ∈ I V α . Thelatter means that there is a partition of unity f β : X → [0 , β ∈ J so that theopen covering f − (]0 , β ∈ J is locally finite and refines the given covering V α , α ∈ I . According to Milnor’s construction of the classifying space BG = G ⋆ G ⋆ G ⋆ . . . as a countable join [9], together with Dold’s analysis ([5], Section 7 and 8), theisomorphism classes of numerable bundles correspond to the homotopy classes ofcontinuous maps Y → BG . We conclude: Corollary 3.4.
The only principal G -bundles over X crs that are numerable are thetrivial ones. Remark 3.5.
Non-numerable principal Z -bundles based on a construction with thelong line appear in [3]. A non-numerable principal R -bundle over a non-Hausdorffspace is sketched in [12], page 350. Remark 3.6.
The results in this section hold true if one uses a simpler space,obtained by attaching only 1-cells e n and no 2-cells, rather than pairs of 1-cells e ± n and 2-cells e n . Of course, the coarser topology is defined in the same way. References [1] M. Artin: On the joins of Hensel rings. Advances in Math. 7 (1971), 282–296.[2] G. Bredon: Sheaf theory. McGraw-Hill Book Co., New York-Toronto-London, 1967.[3] G. Bredon: A space for which H ( X ; Z ) [ X, S ]. Proc. Amer. Math. Soc. 19 (1968),396–398.[4] J. Dieudonn´e: Une g´en´eralisation des espaces compacts. J. Math. Pures Appl. 23 (1944),65–76.[5] A. Dold: Partitions of unity in the theory of fibrations. Ann. of Math. 78 (1963), 223–255.[6] R. Godement: Topologie alg´ebrique et th´eorie des faisceaux. Hermann, Paris, 1964.[7] A. Grothendieck: Sur quelques points d’alg`ebre homologique. Tohoku Math. J. 9 (1957),119–221.[8] A. Grothendieck et al.: Cohomologie locale des faisceaux coh´erent et th´eor`emes de Lef-schetz locaux et globaux. North-Holland, Amsterdam, 1968.[9] J. Milnor: Construction of universal bundles. II. Ann. of Math. 63 (1956), 430–436.[10] H. Miyazaki: The paracompactness of CW-complexes. Tohoku Math. J. 4, (1952), 309–313.[11] L. Steen, J. Seebach: Counterexamples in topology. Second edition. Springer, New York-Heidelberg, 1978.[12] T. tom Dieck: Algebraic topology. European Mathematical Society, Z¨urich, 2008. Mathematisches Institut, Heinrich-Heine-Universit¨at, 40204 D¨usseldorf, Germany
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