HOMEWORK 3
(1) The short exact sequence of chain complexes
0 // A∗
i // B∗
j
// C∗
// 0
iduces the following sequence of homology groups
Hn(A)
i // Hn(B)
j
// Hn(C)
∂ // Hn−1(A)
i // · · · .
Prove that the sequence is exact in the term Hn(A).
(2) Prove that the sequence above is exact in the term Hn(C).
(3) Compute the simplicial homology of S2
. It is a border of the standard 3–
simplex i.e.
S2
∼ [{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}]
(4) Compute the homology groups of Klein bottle or Real projective plane RP2
.
S
T
a a
b
b
V
VV
V
c
Figure 1. Model of Klein bottle.
S
T
a a
b
b
V
VW
W
c
Figure 2. Model of Real projective plane
Date: March 6, 2013.
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