[ALGTOP-L] covering dimension and Cech cohomology

Claude Schochet ad5233 at wayne.edu
Wed Dec 8 13:40:31 EST 2010


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Chris Phillips asks the following question. 

 

Can you suggest references for the connection between covering dimension and
Cech cohomology for compact metric spaces?

 

Here is a rough idea of the kind of question I have.

 

Suppose X is a compact metric space, and suppose I have an open cover which
witnesses the fact that dim (X) > d, that is, there is no refinement of the
cover or order d or less. I would like a way of writing down a Cech cocycle
associated to the orginal cover whose class in H^d (Y) for some subset Y of
X is nontrivial (preferably in fact of infinite order). Actually, it need
not be H^d; H^{d - 1} or H^{d - 2} would do. (In the cases I am interested
in, X is in fact infinite dimensional, and I want to see what happens when I
take the join of the inverse images of a single cover under various maps
from X to X. What I assume as known is that the smallest order of a
refinement grows linearly with the number of maps used.

I want a way to construct higher dimensional nontrivial Cech cohomology
classes from classes of specific degree associated with the original cover.)

 

I recommended Nagata, Modern Dimension Theory, to him as a general
reference.  

 

I will forward responses to him. Thank you!

 

Claude

 

      

 


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vlink=3Dpurple><div class=3DWordSection1><p class=3DMsoNormal><span =
style=3D'font-size:11.0pt;font-family:"Calibri","sans-serif"'>Chris =
Phillips asks the following question. <o:p></o:p></span></p><p =
class=3DMsoPlainText><o:p>&nbsp;</o:p></p><p class=3DMsoPlainText>Can =
you suggest references for the connection between covering dimension and =
Cech cohomology for compact metric spaces?<o:p></o:p></p><p =
class=3DMsoPlainText><o:p>&nbsp;</o:p></p><p class=3DMsoPlainText>Here =
is a rough idea of the kind of question I have.<o:p></o:p></p><p =
class=3DMsoPlainText><o:p>&nbsp;</o:p></p><p =
class=3DMsoPlainText>Suppose X is a compact metric space, and suppose I =
have an open cover which witnesses the fact that dim (X) &gt; d, that =
is, there is no refinement of the cover or order d or less. I would like =
a way of writing down a Cech cocycle associated to the orginal cover =
whose class in H^d (Y) for some subset Y of X is nontrivial (preferably =
in fact of infinite order). Actually, it need not be H^d; H^{d - 1} or =
H^{d - 2} would do. (In the cases I am interested in, X is in fact =
infinite dimensional, and I want to see what happens when I take the =
join of the inverse images of a single cover under various maps from X =
to X. What I assume as known is that the smallest order of a refinement =
grows linearly with the number of maps used.<o:p></o:p></p><p =
class=3DMsoPlainText>I want a way to construct higher dimensional =
nontrivial Cech cohomology classes from classes of specific degree =
associated with the original cover.)<o:p></o:p></p><p =
class=3DMsoPlainText><o:p>&nbsp;</o:p></p><p class=3DMsoPlainText>I =
recommended Nagata, Modern Dimension Theory, to him as a general =
reference.&nbsp; <o:p></o:p></p><p =
class=3DMsoPlainText><o:p>&nbsp;</o:p></p><p class=3DMsoPlainText>I will =
forward responses to him. Thank you!<o:p></o:p></p><p =
class=3DMsoPlainText><o:p>&nbsp;</o:p></p><p =
class=3DMsoPlainText>Claude<o:p></o:p></p><p =
class=3DMsoPlainText><o:p>&nbsp;</o:p></p><p =
class=3DMsoPlainText>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  <o:p></o:p></p><p =
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</o:p></span></p></div></body></html>
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