Homologically trivial fibre

Let us consider a homotopy fibre sequence of connected spaces $$A\rightarrow B\rightarrow C$$ and let $$K$$ be a fixed field. Assume that the homology $$H_{\ast}(A, K)$$ is trivial and that $$C$$ is a nilpotent space (but not simply connected).

Does $$H_{\ast}(B, K)\rightarrow H_{\ast}(C, K)$$ have to be an isomorphism ?

• Yes, by the Serre spectral sequence. – Fernando Muro Jan 29 at 23:03
• Using infinity-categorical language this become very clean (and require no spectral sequences, which in this degenerate cases I consider as a good thing). The fibration is classified by a functor $C\to \mathcal{S}$ classifying the fibers. The assumption tells us that the projection to the constant functor on a point is an $H_*(-;K)$-equivalence and so it is an equivalence on the homology of the colimit, since homology (i.e. tensoring with $K$) is colimit preserving. – S. carmeli Jan 30 at 13:17
• @FernandoMuro It will be nice if you transform your comment to an answer with more details . – TTip Jan 31 at 11:18
• @S.carmeli I would love to understand your comment, but I'm not used to the infinity categorical language... – TTip Jan 31 at 11:19
• @TTip thats ok, ill try to explain. In infinity category theory, a topological space $X$ is a legitimate category, in which points are objects and path are morphisms, and homotopies are "morphisms between morphisms". In this language, there is an equivalence between the homotopy category of fibrations over $X$ and the category of functors from the category $X$ to the infinity category of (nice enough) topological spaces. This associattion takes a fibration to the "functor" of the fibers of the fibration. The total space of the fibration then realizes as the colimit of this functor. – S. carmeli Jan 31 at 12:00

In your situation, the Serre spectral sequence looks as follows

$H_p(C,H_q(A,K))\Rightarrow H_{p+q}(B,K).$

The left hand side is the $$E^2$$ term. There, $$H_q(A,K)$$ carries an action of $$\pi_1(C)$$ induced by the fibration, and $$H_p(C,H_q(A,K))$$ is the homology with local coefficients. Now, if $$A$$ has the homology of a point then then $$H_q(A,K)=0$$ for $$q>0$$ and $$H_0(A,K)=K$$ with the trivial action of $$\pi_1(C)$$ (any self-map of a connected space induces the identity on $$H_0$$). Therefore $$E^2_{p,q}=0$$ for $$q>0$$ and $$E^2_{n,0}=H_n(C,K)$$ is the ordinary homology. Moreover, the spectral sequence ends in the second step and the edge morphism $$H_n(B,K)\twoheadrightarrow E^\infty_{n,0}\subset E^2_{n,0}=H_n(C,K)$$, which is induced by the map $$B\rightarrow C$$, is an isomorphism.

Here is a compromise between Fernando's answer and S. Carmeli's comment. Also, we may as well use $$E_*$$ any homology theory and there's no need to assume $$C$$ is nilpotent, just that every fiber is $$E_*$$-acyclic (so, e.g., if $$C$$ is connected and you have your assumption then we're okay.)

The statement is okay with filtered colimits and equivalences, so we may as well assume $$C$$ is a finite dimensional CW-complex and induct on the dimension. For the inductive step, observe that $$\mathrm{sk}_{n-1}C \to \mathrm{sk}_nC$$ is obtained by cobase change from $$\coprod \partial D^n \to \coprod D^n$$, and it follows that $$B\vert_{\mathrm{sk}_{n-1}C} \to B\vert_{\mathrm{sk}_{n}C}$$ is obtained by cobase change along $$\coprod(B\vert_{\partial D^n} \to B\vert_{D^n})$$. These two pushout diagrams are homotopy pushouts, and the projection map from one to the other is an $$E_*$$-equivalence on the cospans by induction and the assumption on the fiber since $$B\vert_{D^n} \to D^n$$ is equivalent to $$A \to \bullet$$ after trivializing the fibration.

(This is a compromise of S. Carmeli's answer because you can compute the `homotopy colimit over $$C$$' by decomposing $$C$$ along a cell diagram and computing the colimit one piece at a time; this is a compromise of Fernando's answer since one often constructs the Serre spectral sequence using the above pushout diagrams anyway.)

Yet another rephrasing. Homology/chains can be computed in terms of the functoriality of local systems of complexes. In this language, any homotopy type $$C$$ has a constant local system $$k_C$$, and the chains $$C_{*}(C,k)$$ are computed as $$p_!(k_C)$$, where $$p_!$$ is the left adjoint to the pullback functor of local systems for $$p: C \rightarrow *$$. The important, relevant fact is that local systems satisfy base-change for homotopy fibre products, such as your fibre sequence. For connected base $$C$$, your hypothesis that the homology of the fibre is trivial is equivalent to the hypothesis that the canonical map $$f_{!}(k_B) \rightarrow k_C$$ of local systems is an isomorphism, since the fiber of $$f_{!}(k_B)$$ is just $$k$$, by base-change and the hypothesis on the homology of the fibre $$A$$. Now apply $$p_{!}$$ to $$f_{!}(k_B) \rightarrow k_C$$ to obtain the isomorphism $$C_{*}(B,k) \rightarrow C_{*}(C,k)$$.

(Remarks. Really, such an argument has to take place $$\infty$$-category land, or at least in derived categories. Also, the same argument works for any other homology theory, by changing the system of coefficients.)