- #1
Bacle
- 662
- 1
Hi, All:
Just curious to know if there is an interpretation for lower cohomology that is as
"nice", as that of the lower fundamental groups, i.e., Pi_0(X) =0 if X is path-connected
(continuous maps from S^0:={-1,1} into a space X are constant), and Pi_1(X)=0 if
X is path-connected + simply-connected. Are there similarly-nice interpretations
for Cohomology groups, i.e., what can we know about a space X if we
know that H^0(X)=0 , and/or if H_1(X)=0 ? I am aware of Hurewicz' Theorem
(hip, hip Hurewicz) , and of Poincare Duality, but Ii don't see how to get a nice
geometric picture from this. Any Ideas/Suggestions?
Thanks.
Just curious to know if there is an interpretation for lower cohomology that is as
"nice", as that of the lower fundamental groups, i.e., Pi_0(X) =0 if X is path-connected
(continuous maps from S^0:={-1,1} into a space X are constant), and Pi_1(X)=0 if
X is path-connected + simply-connected. Are there similarly-nice interpretations
for Cohomology groups, i.e., what can we know about a space X if we
know that H^0(X)=0 , and/or if H_1(X)=0 ? I am aware of Hurewicz' Theorem
(hip, hip Hurewicz) , and of Poincare Duality, but Ii don't see how to get a nice
geometric picture from this. Any Ideas/Suggestions?
Thanks.