and
be two compact, connected
topological manifolds. Show that the product
is
orientiable if and only if
and
are orientable. Hint: consider
.
be the real projective plane,
be the unit
circle and
the two dimensional torus. Use your
knowledge of the homology of
and
to compute the homology of
the following products:
and
.
be an open set of
. Suppose that
is the union of
convex open sets of
. Show that
the homology groups of
(singular, with integral coefficients) are
finitely generated groups. Hint: use induction on
, the
Mayer-Vietoris exact sequence and the following fact: if
is an exact sequence of abelian groups where
and
are finitely
generated, then
is finitely generated.
of a
topological space X to be the quotient of the product
obtained by identifying
and
to
points. (You may visualize
as the union of two cones with
as a common base.) Show that there is a natural isomorphism
. Hint: let
and
be the points of
corresponding to
and
respectively. Observe that the open sets
and
are acyclic, because a deformation
retraction of
onto
gives, by passing to the
quotients, a deformation retraction of
onto
. What
does ``natural'' mean?
. Embed
as a linear subspace of
. Compute
.
More generally, let
be a closed subspace of
homeomorphic to
. Compute
.
Hint: you may use the fact that for any embedding
of
into
we have
.
be the torus. We regard
as a differentiable
oriented manifold. What are its De Rham cohomology groups? We have a
quotient map
. We use ``periodic''
coordinates
. Thus a smooth function
on
is a smooth
function on
which factors through
, that is, a
doubly periodic smooth function
on
: for all
There are unique differential 1-forms on
denoted
and
whose inverse images under
are the usual
and
on
. Differential forms of degree 1 and 2 on
can be
written uniquely in the form:
where
,
,
are doubly periodic. The integral of
for a suitable orientation is
Likewise, if
is a smooth 1-simlex in
then
is a smooth 1-simplex
in
and:
This being so, show that for any 2-form
on
there is
a form
of degree 1 and a real number
such that
Likewise show that for any closed 1-form
on
there is
a smooth function
on
and real numbers
and
such that
be a connected topological
-manifold. Suppose
that
is not orientable. Show that there is a two-fold cover
where
is a connected orientable topological manifold.
Hint: choose for
the orientation bundle, that is, the disjoint
union over
of the sets of generators of the broups
. Use this to prove that
has a subgroup of index two.
there is an
arcwise connected topological space
such that
. Show
that if
is abelian then there is a connected topological space
such that
.