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Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the nerve of $\mathcal{U}$. I am going to describe two maps from the singular cohomology of $N$ to the singular cohomology of $X$ (with real coefficients). My question is: do these maps coincide? I would be very surprised if they don't, and in fact I think that I could carry through a complete proof of this fact. Nevertheless, the only ideas that came to my mind would probably lead to a rather painful and not very instructive proof, so I was wondering if there could be any result in the literature (or any clever idea) that could help me avoiding unnecessary calculations.

The first map is well-known: Any partition of unity $\{\rho_i\}_{i\in I}$ subordinate to $\mathcal{U}$ induces a map $f\colon X\to N$ defined by $f(x)=\sum_{i\in I} \rho_i(x)\cdot i$. Distinct partitions of unity induce homotopic maps, so $f$ induces a well-defined map $$ f^*\colon H^*(N)\to H^*(X) $$ which does not depend on $\{\rho_i\}_{i\in I}$.

The second map makes use of a double complex. First of all, for every topological space $Z$ let us denote by $C^n(Z)$ the module of real singular cochains on $Z$. When $U$ varies among the open subsets of $X$, the association $U\mapsto C^n(U)$ defines a presheaf on $X$, and we denote by $\mathcal{C}^n$ the associated sheaf, so that $\mathcal{C}^n(U)$ is the space of sections of $\mathcal{C}^n$ over $U$. Since $X$ is nice enough, the inclusion of complexes $C^*(X)\to \mathcal{C}^n(X)$ induces an isomorphism $$ \psi\colon H^*(X)\to H^*(\mathcal{C}^*(X))\ . $$ (Here and below we are using that the usual differential on singular cochains induces a differential on the sections of the associated sheaf).

Let us now denote by $C^{p,q}(X)$ the double Cech-singular complex defined as follows: $$C^{p,q}(X)=\prod_{i_0<\ldots<i_p} \mathcal{C}^q(U_{i_0}\cap\ldots\cap U_{i_p})$$ (so we are using the sheaf rather than the presheaf of singular cochains), where the horizontal differential $C^{p,q}(X)\to C^{p+1,q}(X)$ is the usual Cech differential, and $C^{p,q}(X)\to C^{p,q+1}(X)$ is the differential induced by the singular differential.

After adding the column $C^{-1,q}(X)=\mathcal{C}(X)$ with the obvious restriction map $C^{-1,q}(X)\to C^{0,q}(X)$, the lines of this double complex become exact. Therefore, there exists an isomorphism $$ \varphi\colon H^*(\mathcal{C}^*(X))\to H^*(C^{*,*}(X))\ , $$ where $H^*(C^{*,*}(X))$ denotes the cohomology of the double complex.

On the other hand, since the intersection of $(p+1)$ elements of $\mathcal{U}$ corresponds to a $p$-simplex in $N$, the inclusion of constant maps into $0$-cochains induces an inclusion $C^p_{simpl}(N)\to C^{p,0}(X)$, where we denote by $C^p_{simpl}$ the space of simplicial cochains. After composing with the canonical isomorphism $H^*(N)\cong H^*_{simpl}(N)$, this inclusion induces a map $$ j\colon H^*(N)\to H^*(C^{*,*}(X))\ . $$

I can now formulate my question: do the maps $$ f^*\colon H^*(N)\to H^*(X)\, ,\qquad \psi^{-1}\circ\varphi^{-1}\circ j\colon H^*(N)\to H^*(X) $$ coincide?

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  • $\begingroup$ Why are you sheavifying in the construction of the double complex? I think that the spectral sequences converges to $H^*(X)$ even if you don't do that (also I find it funny that I never heard of the "well-known" first map and I consider the second map a fairly natural one arising from the Mayer-Vietoris spectral sequence) $\endgroup$ Apr 14, 2016 at 15:21
  • $\begingroup$ Hi Denis! The reasons why I am taking sheaves rather than presheaves are two. The first one is that this question arises from the proof of the theorem on amenable coverings in the paper "Foundations of the theory of bounded cohomology", where the problem was posed exactly in these terms. The second one is that without sheafifying the rows of the double complex are not exact, so you need indeed a spectral sequence rather than simply a double complex, and since I am not at ease with spectral sequences I was quite happy about that. $\endgroup$ Apr 15, 2016 at 8:15
  • $\begingroup$ If you have a nice reference for the first map, please tell me. I am using it in a different context to smooth out a piecewise smooth object defined on the nerve. $\endgroup$ Apr 15, 2016 at 10:13
  • $\begingroup$ A reference for the map I was mentioning is e.g. Hatcher's book "Algebraic Topoology", Section 4.G (the book is available here: math.cornell.edu/~hatcher/AT/AT.pdf ). I also found a reference to this map in Spanier's book "Algebraic topology", in the exerxises at page 152. $\endgroup$ Apr 18, 2016 at 9:31

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