**Week 8**

1. An *n*-dimensional *topological manifold-with-boundary*
is a space locally homeomorphic to R^*n*- (the set of points in
R^*n* with *x*_1 < = 0).
The *boundary* D*M* is the set
of points corresponding to *x*1 = 0 under one (and therefore any)
of those local homeomorphisms. A smooth atlas on such a
manifold is a set of coordinate charts of this type
which are differentiably
related. The only difference from the no-boundary definition
occurs with the definition of D*y*_*i*/D*x*_1
at a point with *x*_1 = 0,
where the *x*'s and *y*'s are coordinates in two systems overlapping
at a boundary point. We take this to mean the one-sided derivative,
taken from the side where *x*_1 < = 0.

The boundary D*M* is an (*n*-1)-dimensional manifold: it inherits
a smooth atlas from *M*, given by the restriction to D*M* of
the special smooth coordinate systems defining *M* as a
smooth manifold-with-boundary. Since *y*_1 and *x*_1 are constant
along the boundary, the functions (*y*_2,...,*y*_*n*)
and (*x*_2,...,*x*_*n*)
give smoothly related (*n*-1)-dimensional coordinates on D*M*.

Suppose *M* is oriented: an atlas has been chosen where all
overlapping coordinates are positively related (the Jacobian
matrix of the change-of-coordinates map has positive determinant).
This also makes sense for a manifold-with-boundary; furthermore,
if we assume as above that the boundary is given by *x*_1 = 0 (and the
rest by *x*_1 < 0) in those
coordinate charts intersecting the boundary, then when two such
charts intersect the Jacobian
D(*y*_2,...,*y*_*n*)/D(*x*_2,...,*x*_*n*)
will be positive. So D*M* inherits an orientation from *M*.

*Examples.*

1. *M* = [*a*,*b*] with *a* < *b*,
a closed interval. This is clearly
a 1-dimensional manifold-with-boundary. A boundary-defining
coordinate near *b* is *h*_1(*x*) = *x* - *b* ;
a boundary-defining
coordinate near *a* is *k*_1(*x*) = *a* - *x*.
These two coordinates
are *not* positively related at an interior *x*.
To study the orientation induced on the boundary, we make
the convention that an orientation on a 0-dimensional
manifold is an assignment of a sign, + or -, to each of
its points; and that a boundary point receives the +
orientation if it is defined as above by a *positive*
boundary-defining coordinate, and - otherwise.
Giving *M* the
orientation inherited from the *x*-axis makes the coordinate
*h*_1 positively oriented, and the coordinate *k*_1
negatively. Consequently D([*a*,*b*]) = {*b*} - {*a*}.

2. *M* = *D*^2, the disc {*x*^2 + *y*^2 <= 1}.
The boundary
is the circle *S*^1. Give *M* the orientation inherited from
the standard *x*,*y* orientation of the plane. Then near the
point (1,0) a positive, boundary defining coordinate system
is, for example, *h*_1(*x*,*y*) =
(*x*/sqrt(1-*y*^2) - 1), *h*_2(*x*,*y*)=*y*;
so that *h*_1 <= 0 means *x* <= sqrt(1-*y*^2) as desired.
The orientation induced on *S*^1 near (1,0) is that defined
by increasing *h*_2, i.e. increasing *y*. This gives *S*^1 the
counter-clockwise orientation.

**Stokes' Theorem.** With *M*, D*M* as above, consider a smooth
(*n*-1)-form *omega* on M. The integral of d *omega*
over *M* is equal to the integral of *omega* over D*M*.

/ / | domega= |omega. /M/DM

2. *Differential forms and topology.*
Write *Omega*^*p*(*M*)
for the (R-)vector-space of differential
forms of degree *p* on the *m*-dimensional manifold *M*. This
space is 0 for *p*> *m*.
Since dd = 0 we can construct in each dimension < = *m* the
vector-space *H*^*p*(*M*) =
*H*^*p* = *Z*^*p*/*B*^*p* =
(ker d)/(im d), where
*Z*^*p* = ker d is the kernel of d: *Omega*^*p*
--> *Omega*^(*p*+1)
and B^*p* = im d is the image of d: *Omega*^(*p*-1) -->
*Omega*^*p*. The elements of these sub-vector-spaces
are called *p*-cocycles and *p*-coboundaries, respectively;
while *H*^*p*(*M*) is called the *p*-th
de Rham cohomology vector-space
of *M*.

*Examples.* At this point we can calculate only a few
examples of de Rham cohomology: for example *M* = {*} (a point)
and *M* = *S*^1 (the circle).

*M* = {*} is a 0-dimensional manifold. The only non-zero
*Omega*(*M*) is consequently *Omega*^0(*M*), the
vector-space of real-valued functions; here functions on
a point. This vector-space is clearly R. Since there are
no other non-zero *Omega*(*M*)'s, *Z*^0 = *Omega*^0
and *B*^0 = 0; consequently *H*^0 = R, and all others are 0.

The circle is 1-dimensional, so there are only non-zero
*Omega*^0 and *Omega*^1. Clearly *B*^0 = 0 and
*Z*^1 = *Omega*^1. An element of *Omega*^0,
i.e. a real-valued function *f*(*theta*), is in *Z*^0
if d*f* = 0, so *f* is a constant, and the vector-space
*H*^0 is again R. To calculate *H*^1, consider the map
*I*: *Z*^1 = *Omega*^1 --> R which assigns to each 1-form
*f*(*theta*)d(*theta*) its integral.
* Class exercise* ker *I* = *B*^1. Consequently
*H*^1 = *Z*^1/*B*^1 = *Z*^1/(ker *I*) = R.

The rest of the class was spent on the definitions
of cochain complex, exact sequence, and homomorphism
of cochain complexes. Examples: for a smooth manifold
the sequence

d d d 0 -->is a cochain complex (theOmega^0 -->Omega^1 --> ... -->Omega^n--> 0

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