Math 726
Fall 2002
Homework
4
Due
on the last day of class. Do not forget to mark your “favorite” problem.
1.
Compute
the all of the homology groups for the following space:
b
a a
b
2.
a)
Compute all the homology group of the 5-fold dunce cap:
a a
a a
a
b) Compute the homology of the analogously defined k-fold dunce
cap.
3.
Let
be a chain map. We
define
and we define
by
We call this sequence
of modules and maps
(the mapping cylinder of
).
a)
Show
that is a complex.
b)
If
is a complex, let
obtained by
increasing all indices by 1. Show that
c)
If
is a chain map, then
show there is an exact sequence of complexes:
d)
Show
there is a long exact sequence:
where is induced by
e)
Show
that is an isomorphism for
all
if and only if
is acyclic (that is
forms an exact
sequence).