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Geometric Interpretation of (lower) Cohomology?

  1. Sep 16, 2011 #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.
     
  2. jcsd
  3. Sep 16, 2011 #2

    quasar987

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    By my recollection, pi_0(X) is defined as the set of connected components. (not path conencted).

    On the other hand, H_0(X;Z) is canonically isomorphic to the free abelian group on the set of path-connected components of X. So H_0 (or rather its rank) counts the # of connected components. (In particular, H_0 = 0 is impossible). And since H^0(X;Z) is always isomorphic to Hom(H_0(X;Z),Z), its rank is the same as that of H_0, so it too counts path-connected components and H^0 = 0 is impossible.
     
  4. Sep 16, 2011 #3
    Quasar: By 0 I meant the trivial group, i.e., the group with 1 element.

    And I think Pi_0(X)=0 counts the number of maps , up to homotopy from

    S^0:={-1,1} into X ; so we want the image of f to contract into a constant

    c, so for Pi_0 to be 0, X must be path connected. But there is too the fact that

    for a map f into a space X to be trivial, X must be connected, since, I think that

    contractible spaces are connected. Wikipedia says Pi_0(X)==0 iff X is path-connected.

    But then path-connected implies connected, so I'm not sure.
     
  5. Sep 16, 2011 #4

    quasar987

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    For some reason I was interpreting "0" as "void", sorry.
     
  6. Sep 17, 2011 #5
    Bacle- you are right about pi_0. pi_n can be associated with homotopy classes of maps from S^n into the space in question, so pi_0 counts path components.

    Different homology/cohomology theories describe different things. For example, in normal homology/cohomology (e.g. simplicial, singular, etc.) H^0 again describes the number of path components (check this yourself- it's not too hard to prove!). However, there are other theories which tell you something different, for example, Cech cohomology tells you the number of connected components (not path components) [at least, I'm pretty sure of this].

    For example, if your space is the topologists sine curve, then if you throw singular cohomology at it, then the zero group will be Z+Z telling you that there are two path components. Use Cech cohomology though, and it can't tell that there are two path components (I like to think of it as a sort of "blurry microscope, that can't pick out weird asymptotic behaviour, such as in the topologists sine curve), and it will give you Z saying that there is only one connected component.

    In general, to get an interpretation for all cohomology groups, I'm sure you've probably seen the theorem that elements of the singular cohomology groups can be related bijectively to elements of [X,K(pi,n)], homotopy classes of maps from X into the space K(pi,n) where K(pi,n) is a Eilenberg Maclane space. So for low dimensions, a K(Z,1) is the circle, so H^1 is describing maps of X into the circle. For K(Z,2) we have the infinite projective space- the higher ones get progressively more difficult to visualise though (and the cohomology of these spaces becomes more and more tricky).
     
  7. Sep 17, 2011 #6
    No problem, Quasar; thanks, Jamma.
     
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