Cubic Singular Homology
Now that we’re armed with chains — formal sums — of singular cubes we can use them to come up with a homology theory. Since we will use singular cubes to build it, we call it “cubic singular homology”.
So, for each 
we have the abelian group 
of chains. For each one we want to define a map 
with the property that 
so we can use this sequence of abelian groups as the setup for a homology theory. Now what would be an appropriate name for this sequence of groups of chains? I think “chain complex” fits nicely, no? It’s almost as if that’s where the name came from…
Etymologies aside, we need to pass from a -dimensional chain to a 
-dimensional one. And since every -dimensional chain is a formal sum of singular -cubes, we really just need to define it on -cubes and extend by linearity.
The key idea here is that given a -cube 
we want 
to correspond to the boundary of , and so it will be a certain combination of the -dimensional “faces” of . Heuristically, each face itself has faces, and each of these 
-dimensional faces is part of two of the -dimensional faces, and when we work everything out they will turn out to cancel each other off, leaving an “empty” second boundary.
So for each dimension we’re going to have two faces, one of which we get by setting that component to 
and the other of which we get by setting that component to 
. Explicitly, we’ll define the following faces of the standard cube 
:
for any ![a\in[0,1]^{k-1}](http://s0.wp.com/latex.php?latex=a%5Cin%5B0%2C1%5D%5E%7Bk-1%7D&bg=e6e6e6&fg=333333&s=0 "a\in[0,1]^{k-1}")
. Then for any other singular cube we define the face 
. Then we define the boundary operator by
As an example, if is a -cube then 
. We have to be careful here: this is not a “real” subtraction — 
is a point in 
, remember, which may not have any sense of subtraction at all. This is just a formal subtraction of one -cube — one manifold point — from another. We will extend our definition to -cubes by setting 
for all -cubes , and thus we automatically have 
for -cubes .
More generally for a singular 
-cube , we calculate
Now, if 
then it’s straightforward to check that 
; the 
creeps in because after we insert something between 
and 
we have an extra place to go to insert the other value. Thus we find that 
, and so each term in the big sum above shows up twice. The thing is, one time it shows up with sign 
and the other time it shows up with sign 
, which cancels off the first appearance. And so the whole sum collapses to , just like we asserted.
And so we again define the group 
of closed chains to be those chains 
with 
and the group 
of exact chains to be those where there exists some 
with 
. Again, 
, since . And again, we define the homology group 
.
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August 9, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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