# $S^1$ normal bundle on divisor and Serre spectral sequence

Let $$D\subset X$$ be a smooth divisor in a smooth complex variety. On $$D$$ we have the normal bundle $$N$$. Removing the zero section and retracting we get an $$S^1$$ bundle. Call this bundle $$N'$$. Now I'd like to understand the first homology of $$N'$$. We get from the Serre spectral sequence $$H_i(D,H_j(S^1,\mathbb{C}))\Rightarrow H_{i+j}(N',\mathbb{C}).$$ On this $$E_2$$ page, we get a map $$E^2_{2,0}=H_2(D,\mathbb{C})=\mathbb{C}\to E^2_{0,1} =H_0(D,\mathbb{C})$$. I guess this map is taking the cap product with the Euler class of $$N'$$, i.e. the first Chern class of $$N$$, but I'm not sure about this. Is this correct?

It follows that if the Chern class is non-trivial, then $$E^2_{0,1}=E^\infty_{0,1}=0$$. Since clearly $$E^2_{1,0}=E^\infty_{1,0}=H_1(D,\mathbb{C})$$, we thus find that $$H_1(N',\mathbb{C})\cong H_1(D,\mathbb{C}).$$ Now I wonder if this isomorphism is realised by the natural map? I.e. is the map just the pushforward in homology induced by the projection?

I tried reading up on the construction of the Serre spectral sequence, but it doesn't really help me with understanding the map to be honest.

• Look up the Thom isomorphism. – wnx Aug 23 '19 at 21:21
• If $D$ is the origin in $\mathbb{A}^1$, $N'$ is $S^1$ and you seem to be implying $H_1(S^1,\mathbb{C})=H_1(\mathrm{pt},\mathbb{C})$...? – Mattia Talpo Aug 24 '19 at 15:28
• @MattiaTalpo in this case the first chern class is trivial, and hence the argument does not apply. In fact, the spectral sequence correctly gives the cohomolgy of $S^1$ – user2520938 Aug 24 '19 at 15:35
• Ok, sorry, I had missed that bit of text... – Mattia Talpo Aug 24 '19 at 15:42

This already gives that the projection is an isomorphism on $$H_1$$ when the Euler class is non-trivial. That the projection agrees with the edge homomorphism in the Leray spectral sequence is shown in many books which cover spectral sequences, e.g. Switzer's "Algebraic Topology: Homotopy and Homology", 15.29.
• Thank you for this answer. P.s, we don't need simply connected $D$ right? I think that $H_1(S^1)$ is always the trivial local system on $D$, since the monodromies are degree one maps induced from the transition functions of the normal bundle right? – user2520938 Aug 29 '19 at 21:01