# Fibre transfer of $\mathbb{S}^1$-bundles

Let $$p:E\to X$$ and $$p':E'\to X'$$ be two orientable $$\mathbb{S}^1$$-bundles. Denote their homological transfers by $$p_!:H_*(X)\to H_{*+1}(E)$$ resp. $$p'_!$$.

Now let $$(u,f)$$ be a bundle morphism ($$u:E\to E'$$ and $$f:X\to X'$$ and $$p'\circ u = f\circ p$$) and assume that $$u_x:E_x\to E_x$$ has degree $$1$$ for all $$x\in X$$.

Can we conclude that $$u_*\circ p_!=p'_!\circ f_*$$ as maps in homology?

Edit: I think I solved it, and it immediately generalises to $$\mathbb{S}^k$$-bundles:

1. Let $$q:(D,E)\to X$$ be the associated $$(\mathbb{D}^2,\mathbb{S}^1)$$-bundle, $$v:(D,E)\to (D',E')$$ the extended morphism and $$\tau\in H^2(D,E)$$ the Thom class. Our condition tells us that $$\tau=v^*\tau'$$.
2. Consider the connecting homomorphism $$\delta_*:H_*(D,E)\to H_{*-1}(E)$$ resp. $$\delta_*'$$. By naturality $$u_*\circ \delta_* = \delta'_*\circ v_*$$.
3. We have the Thom isomorphism $$t:H_*(D,E)\to H_{*-2}(X),x\mapsto q_*(\tau\frown x)$$ and $$t'$$. We see that $$(t')^{-1}\circ f_* = v_*\circ t^{-1}$$ by $$(f_*\circ t)(x) = f_*q_*(\tau\frown x) = q'_*v_*(v^*\tau'\frown x)=q_*(\tau'\frown v_*x)=(t'\circ v_*)(x)$$
4. In the Gysin sequence, $$p_!$$ appears as $$p_!=\delta^*\circ t^{-1}$$. Thus, finally $$u_*\circ p_! = u_*\circ \delta_*\circ t^{-1} = \delta'_*\circ v_*\circ t^{-1} = \delta'_*\circ (t')^{-1}\circ f_* = p'_!\circ f_*.$$
• This is obvious for trivial bundles, and the general case should follow by a Mayer-Vietoris argument. – ThiKu Mar 21 '19 at 9:36
• What you denote transfer should really be called Gysin map (or "umkehr map"), not transfer. – Jens Reinhold Mar 21 '19 at 15:01
• Hm … what is the difference? I have several definitions for $p_!$: One appears in the homological Gysin sequence (and is the inverse Thom isomorphism followed by the connecting homomorphism of a relative sequence for the $(\mathbb{D}^2,\mathbb{S}^1)$-bundle $(D,E)$), one comes from the Leray spectral sequence and one via Poincaré duality if $X$ is a closed and oriented manifold. Are they all the same? Which one is “transfer”, which one “umkehr map” and which one “Gysin map”? – FKranhold Mar 21 '19 at 15:20