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).