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TOPOLOGY II
ASSIGNMENT 5 (SIMPLICIAL HOMOLOGY)
Problem 5.1. Consider the chain complex
0 # 4
-# Z # 3
-# Z # 2
-# Z # 1
-# 0
where # 3 (z) = 2z and # 2 (z) = 0. Compute its homology.
Problem 5.2. Compute the homology of the singleton and the segment [0, 1].
Problem 5.3. Compute the homology of the subdivided segment
Problem 5.4. Compute the homology of the boundary of the triangle and the boundary of the
square. Compare.
Problem 5.5. Compute the homology of the boundary of the tetrahedron and the boundary
of the cube (its faces are triangulated by means of their diagonals). Compare.
Problem 5.6. Prove that the induced homomorphism is well defined.
Problem 5.7. Prove the Poincar’e lemma in details (check that the two simplices
[v 0 , . . . , –
v j , . . . , –
v i , . . . , v n ] do appear with opposite signs).
Problem 5.8. Prove that a simplicial space X is connected if and only if H 0 X = Z.
Problem 5.9. Compute the homology groups H # (M ˜
ob, Z) of the M˜obius strip.