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Monday, July 20, 2020

Cohomology: Mayers-Vietoris sequence.

Let U,V be an open cover of Manifold M. Remember it is very much possible that UV.
Define the following inclusion maps
iu:UM,iu(p)=piv:VM,iv(p)=pjU:UVU,ju(p)=pjV:UVV,jv(p)=p
These inclusion map iU(p)=p from U to M and similar map iv(q)=q from V to M etc.,

Based on these inclusion maps one can define pull back maps of differentials
iU:Ωk(M)Ωk(U)
Similarly one can define a pull back for iV
iV:Ωk(M)Ωk(V)
 Similar pull back maps are defined for jU,jv.
 jU:Ωk(UV)Ωk(U)jV:Ωk(UV)Ωk(V)

 

 By restricting to U and to V, we get a homorphism of vector spaces
 i:Ωk(M)Ωk(U)Ωk(V)

 defined via
 σ(iUσ,iVσ)

 Using this, define the difference map,
 j:Ωk(U)Ωk(V)Ωk(UV)
 by j(ω,τ)=τω.

 This map indicates what to do with common vectors that belong to both U and V. Similar maps are used in Finite dimensional vector spaces to prove dimensionality theorem when U,V are subspaces whose intersection is non-empty.

 Here τ,ω are pull backs maps shown before.
 ω=jUωτ=jvτ

 j is a zero map when UV=.

 Proposition
 For each integer k0, the sequence
 0Ωk(M)iΩk(U)Ωk(V)jΩ(UV)0
 is exact.

 Proof:
 To show that this exact sequence, we need to show at each node image of previous function to this node is same as kernel from this node to next node.
 We will start with first node - Ωk(M).
 0 vector maps every function to 0 in the Ωk(M) which is in kernel of i. Hence, im(0Ωk(M))=keri.
 
 To prove exactness at Ωk(UV), we need to show that j is surjective or onto as next maps takes everything to zero. Thus kernel of next map is all of Ωk(UV) which is range of j.
 
 We are already given j map in the previous section. This map, together with a very nice partitions of unity, helps us to establish the onto of j map.

 say ωΩk(UV). Let pU,pv be functions that form partitions of unity. Define
pUω={pvω when ,xUV0 otherwise ,xU(UV)pVω={pUω when ,xUU0 otherwise ,xV(UV)
The niceness of partition of unity allows the following to happen.
j(puω,pvω)=pvω+puω=ω on UV
This shows that j is onto and from the fact the next function sends everything to 0, we have proved that this is a short exact sequence.

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