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Topic: Top Homology of Manifold with Boundary is Zero. True? Why?
Replies: 2   Last Post: May 28, 2013 11:54 PM

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W. Dale Hall

Posts: 71
Registered: 2/11/05
Re: Top Homology of Manifold with Boundary is Zero. True? Why?
Posted: May 14, 2013 10:41 PM
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baclesback@gmail.com wrote:
>
> Hi, All:
>
> I'm trying to see if it is true that the top homology of
>
> an n-manifold with boundary is zero. I have tried some examples;
>
> a closed annulus ( homotopic to S^1 , so H^2(S^1)=0 , R^n with
>
> a boundary copy of some R^m m<n retracts to R^m , etc.)
>
> It seems strange that a manifold would "lose its orientability"
>
> if we capped-in a boundary. It seems like this boundary would bound
>
> all the n-cycles that had no boundary before the boundary was capped-in;
>
> otherwise, how is the quotient Cycles/Boundaries suddenly zero?
>
> I tried some MAyer-Vietoris, but got nowhere. Any Ideas?
>
> Thanks.
>


The n-dimensional homology of an orientable, *connected* n-dimensional
manifold with boundary vanishes.

Note that the n-dimensional homology of a connected, closed,
orientable n-manifold is Z. If you remove a small disc to yield the
manifold-with-boundary Mo, then the Mayer-Vietoris sequence for
M = Mo U D^n looks like this in dimensions n & n-1:

(this needs to be viewed in a fixed-spacing font to make sense)

H_n(Mo)
H_n(S^(n-1)) --> (+) ---> H_n(M) --,
H_n(D^n) /
/
/
__________________________________/
\ (boundary)
\ H_(n-1)(Mo)
`--> H_(n-1)(S^(n-1)) --> (+) --> H_(n-1)(M) -->...
H_(n-1)(D^n)

Plugging in what values we know, we get this:


H_n(Mo)
0 --> (+) ---> Z --------,
0 /
/
/
__________________________________/
\ (boundary)
\ H_(n-1)(Mo)
`----------> Z ----------> (+) --> H_(n-1)(M) -->...
0

or
(bdy)
0 --> H_n(Mo) --> Z --> Z --> H_(n-1)(Mo) --> H_(n-1)(M) --> ..

Next, note that H_n(Mo) can't map into a proper subgroup of H_n(M),
since the next homomorphism is into Z (the quotient group would have
nonzero torsion, which doesn't exist in Z). That is, the only options
are H_n(Mo) = 0, or H_n(Mo) maps *onto* H_n(M).

Further, the homomorphism

H_(n-1)(S^(n-1)) ---> H_(n-1)(Mo)

must be zero, since S^(n-1) bounds in Mo (after all, it's the boundary
of Mo). What we find, therefore, is that H_n(M) maps isomorphically
onto H_(n-1)(S^(n-1)).

Your query

It seems strange that a manifold would "lose its
orientability" if we capped-in a boundary. It seems
like this boundary would bound all the n-cycles that
had no boundary before the boundary was capped-in;
otherwise, how is the quotient Cycles/Boundaries
suddenly zero?

seems a bit confused.

In the first place, orientability isn't the issue. The issue is
in finding n-dimensional cycles (i.e., chains with 0 boundary). For a
connected orientable manifold, the fundamental class (that is,
the chain that carries n-dimensional homology) *is*, in some sense,
the full manifold. If it's simplicial homology, and your manifold
is representable as a simplicial complex, then the class [M] can
be represented as the sum of all top-dimensional simplices (with
appropriate orientations). No proper subset of simplices will add up
to a chain whose boundary vanishes.

Further, I don't understand the phrase "if we capped-in a boundary".
A manifold with boundary (for lack of better phraseology) actually
*has a boundary*. Any sum of n-simplices will have nonzero boundary,
so won't be cycles at all. The boundary operator from C_n(Mo) to
C_(n-1)(Mo) has zero kernel. There are *no* n-dimensional cycles
to begin with. Nothing suddenly becomes zero, since there was never
anything there to begin with.

For instance, take one of your examples, say, the annulus. Have a
look at what the 2-chains represent geometrically. Note that you
simply can't arrange 2-dimensional simplices in such a way as to
cancel out the boundary. You can force the boundary to reside in
the bounding circles of the annulus, but that's the best you can do.

You might be heartened to note that the *relative* homology,

H_n(M,dM) = Z

for a connected orientable n-manifold M with boundary dM. In this
case, all those n-chains you pushed to force their boundaries into
dM now (in the relative case) become cycles, and you get n-cycles.
Once you have cycles, you can see homology in dimension n.

I hope I've cleared some confusion & not strewn any more of the same.

Dale




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