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INIDEPENDENT UNIVERSITY OF MOSCOW

6. MAYER{VIETORIS SEQUENCE

TOPOLOGY-2, FALL 2012

The following is for the students who solved Problem 4.2 (the Bockstein's exact sequence) or are ready to believe its statement. Let X = A B be a CW-space, A; B X be its CW-subspaces (unions of cells that are CW-spaces) such that A B is a CW-subspace of both A and B . Let uA : A X , uB : B X , wA : A B A and wB : A B B be tautological embeddings. Denote by Z (Y ) the cell complex of a CW-space Y .
Z (A B ) A - B Z (A) Z (B ) uA +uB Z (A B ) 0 is an exact sequence of - complexes. The corresponding exact sequence of cell homology (see Problem 4.2) is called the Mayer{Vietoris sequence. The Mayer{Vietoris sequence for singular homology is a bit more tricky: x a barycentric subdivision of the standard simplex: n = n+1 n; ; here n+1 is the permutation group of 0; 1; : : : ; n, and n; def {(x0 ; : : : ; xn ) = n | x(0) · · · x(n) }. For every x an a ne map wn; : n n; , sending the vertex xi = 1 of n (where i = 0; : : : ; n) to the vertex of n; where xj = 1=(i + 1) if j {(0); : : : ; (i)}, and xj = 0 otherwise. De ne the homomorphism n : Cn (X ) Cn (X ) by n (f ) = n+1 (-1)sign() f w .

Problem 1. Prove that 0



w

(-

w)

Problem 2*. Prove that n is a morphism of complexes, and ( n ) : Hn (X ) Hn (X ) is trivial (here Hn (X ) is the singular homology of X ). A;B Let X = A B where A; B X are open; denote by Cn (X ) the subcomplex of Cn (X ) spanned by singular simplices f : n X where f ( n ) A or f ( n ) B . Problem 3*. (a) Prove that the inclusion : C A;B (X ) C (X ) is a morphism of complexes, and is trivial on homology | hence, the homology of C A;B (X ) is the same as the singular homology of X . (b) Let wA : C (A B ) C (A), wB : C (A B ) C (B ), uA : C (A) C A;B (X ) and uB : C (B ) C A;B (X ) be tautological inclusions. w (-w ) Prove that 0 C (A B ) A - B C (A) C (B ) uA +uB C A;B (X ) 0 is an exact sequence of complexes. The - corresponding exact sequence of singular homology is called the Mayer{Vietoris sequence. Problem 4. Compute the Mayer{Vietoris sequence with (a) X = S n , A and B are the upper and the lower half-sphere, respectively. (b) A and B are two cylinders glued together by the bases to form X = T2 . (c) A and B are two cylinders glued together by the bases to form the Klein bottle. Problem 5. Let X = Y be the suspension. Use the Mayer{Vietoris sequence to express H (X ) via H (Y ). Problem 6. Let X = S 3 , K S 3 be a knot, A be its thin tubular neighbourhood, and B be the closure of X \ A. Compute the Mayer{Vietoris sequence of (X; A; B ) if K is (a) an unknot, (b) a trefoil. Problem 7. Look in Wikipedia (or elsewhere) when Leopold Vietoris was born, and when he died.