Continuity of Alexander-Spanier cohomology

Suppose that a paracompact space $X$ is the inverse limit of paracompact spaces $X_{i}$ (that is $X=\varprojlim X_{i}$) and $H^{\ast }$ is Alexander-Spanier cohomology with closed supports. Then the equality $H^{\ast }\left( X,k\right) =\varinjlim H^{\ast }\left( X_{i},k\right)$ is true? (where $k$ is a field of characteristic zero.)

• Do you mean closed support or compact support? (Every function on a topological space has closed support by definition.) You probably are aware of the fact that continuity holds for Alexander-Spanier cohomology (no support condition) on paracompact spaces, by identification with Cech cohomology (via sheaf cohomology). This is discussed in Spanier's "Algebraic topology". – Matthias Wendt Dec 20 '16 at 13:13
• @MatthiasWendt I know only that the continuity holds for Alexander-Spanier cohomology with compact supports on locally compact spaces. You can look at Bredon's Sheaf Theory book (14.6. Corollary page 103). I dont know that it holds on Alexander-Spanier cohomology with closed support on paracompact spaces. – Mehmet Onat Dec 21 '16 at 14:04
• @MatthiasWendt Actually i want to know that if $G$ is a compact totally disconnected group acting on a paracompact space X, then the orbit map $X\rightarrow X/G$ induces isomorphism (or monomorphism) on Alexander-spanier cohomology with closed supports? – Mehmet Onat Dec 21 '16 at 14:09
• @MatthiasWendt Is it true that Cech cohomology on paracompact spaces has the continuity property? – Mehmet Onat Dec 21 '16 at 14:11
• My comment on continuity of Cech cohomology was a bit hasty. The classical sources only do this for compact spaces. However, there is a weak continuity for paracompact spaces by results of Lee and Raymond. I'll search a bit more through the literature and then come back to give more details. – Matthias Wendt Dec 21 '16 at 19:58

First of all, I am not sure if the below answers the question because of a terminological issue. I only know "closed support" in the context of Borel–Moore homology. For Alexander–Spanier cohomology, being defined in terms of functions where support is by definition closed, "closed support" seems to me to refer to the full complex where all functions are allowed – and this is what I am talking about in the below.

The Alexander–Spanier cohomology can be identified with Čech cohomology. On paracompact spaces, this is proved in Spanier's "Algebraic topology", Chapter 6, Section 8 "Fine presheaves". This actually holds for general spaces by a theorem of Dowker:

• C. Dowker. Homology groups of relations. Ann. Math. 56 (1952), 84–95.

So the continuity question for Alexander–Spanier cohomology is equivalent to the continuity question for Čech cohomology. However, in my comment I was a bit too hasty about the continuity for Čech cohomology. The classical references (e.g. Bredon's "Sheaf theory", Spanier's "Algebraic topology", or the original papers of Steenrod and Spanier) prove continuity for Čech cohomology on the category of compact Hausdorff spaces.

However, Čech cohomology satisfies a weak continuity property for paracompact Hausdorff spaces, by a result of Lee and Raymond:

• C.N. Lee and F. Raymond. Čech extensions of contravariant functors. Trans. Amer. Math. Soc. 133 (1968), 415–434. (paper available here)

Specifically, this is Theorem 5 in their paper. Note, however, that all the spaces in the inverse system are required to be embedded into one big space, with maps given by subspace inclusions. They define nested systems to additionally be those where the inverse limit is given by intersection and where every neighbourhood of the limit in the ambient space contains one of the spaces of the system. Weak continuity gives isomorphisms in cohomology only for inverse limits of nested systems. This is all discussed in Section 4 of the Lee–Raymond paper.

There's also a paper of Watanabe

• T. Watanabe. The continuity axiom and the Čech homology. Geometric topology and shape theory (Dubrovnik, 1986), 221–239, Lecture Notes in Math., 1283, Springer, 1987.

In the paper he discusses at length continuity properties of extensions of cohomology theories. If I'm reading his Corollary 22 right, it says that Čech cohomology is continuous on the category of pairs $(X,A)$ where X is a paracompact Hausdorff space with a closed subset $A$. Note, however, that there Čech cohomology is defined in terms of normal open coverings, i.e., those which have a partition of unity.

• First of all, thank you for your time and references. I looked at T. Watanabe. Example 3 show that compacteness is necessary. It is not true for paracompact spaces, because $X_i$ is paracompact. I think. – Mehmet Onat Dec 26 '16 at 11:38