480 APPENDIX A. HOMOLOGICAL METHODS∗

the usual way. (c,d)+(ĉ, d̂)= c+ ĉ+d + d̂. Then f# is clearly one to one and g# is onto.

Also, if g# (c,d) = 0 then c+d = 0 and so d =−c so (c,d) = (c,−c) ∈ Im( f#). Thus thisis a short exact sequence. We also assume f#,g# are chain maps so ∂ f# = f# (∂ ⊕∂ ) and∂g# = g# (∂ ⊕∂ ) where (∂ ⊕∂ ) does the obvious thing (∂ ⊕∂ )(c,d) = (∂c,∂d). Thus thisyields a short exact sequence of chain complexes. It follows from Theorem A.6.3 that thereexists a long exact sequence of homology groups.

· · · → Hn (U ∩V )f∗→ Hn (U)⊕Hn (V )

g∗→ Hn(SU,V

n (X)) ∆→ Hn−1 (U ∩V )

f∗→ ···

This is called the Mayer Vietoris sequence.Also notice that if h : X→ X̂ is continuous with h(U)⊆ Û ,h(V )⊆ V̂ and X̂ = int

(Û)∪

int(V̂)

then the squares in the following diagram must commute. This is a consequenceof Proposition A.6.5 and the fact that the corresponding squares in the short exact se-quences of chains involving h# commute. Note how f ,g make perfect sense independentof, X ,U,V,Û ,V̂ , X̂ or on h.

→ Hn (U ∩V )f∗→ Hn (U)⊕Hn (V )

g∗→ Hn

(SU,V

n (X))

∆→ Hn−1 (U ∩V )f∗→

↓ h∗ ↓ h∗⊕h∗ ↓ h∗ ↓ h∗

→ Hn(Û ∩V̂

) f∗→ Hn(Û)⊕Hn

(V̂) g∗→ Hn

(SÛ ,V̂

n(X̂)) ∆→ Hn−1

(Û ∩V̂

) f∗→

Lemma A.7.1 For U,V open sets containing X and for h : X → X̂ satisfying h(U) ⊆Û ,h(V ) ⊆ V̂ where X̂ = int

(Û)∪ int

(V̂)

then the above diagram is valid in which therectangles commute.

It is time for examples at long last. We do have a couple of good ones already. Recallthat H0 (X) = Z in case X is path connected. This is from Theorem A.2.5. Also recall thatfrom Proposition A.2.8 Hn (X) is the direct sum of homology groups of the path compo-nents of X . I will refer to Hn

(SU,V

n (X))

as Hn (X) from now on because that material onsubdivisions says that if c is a cycle, we can obtain that it is homologous to one in whichall the simplices are supported in one of U or V .

A.8 The Homology Groups of SpheresThis is done using the Mayer Vietoris sequence and induction which reduces to S1.

Example A.8.1 S1 is the unit circle x2 + y2 = 1. Letting this be X , what are its homologygroups?

xV is everything but the top

U is everything but the bottom

It is certainly path connected so H0(S1)= Z but what of H1

(S1)? Let U be all of S1

other than the bottom point (0,−1) and let V be all of S1 other than the top point(0,1).Hn (U)⊕Hn (V ) = (0,0) because U,V are both homeomorphic to (−1,1) a convex set