Non constant sheaf on a contractible space

Question

while trying to calculate the coohomology (with complex coefficients) of a bundle over a simply connected base, I met the following

PROBLEM

Suppose you have a space $X$ which contracts on $x \in X$, and a sheaf $F$ on $X$ such that $F_p \simeq V$ for all $p \in X$, where $V$ is a $\mathbb{C}$ vector space of finite dimension. Is it true that $F$ is the constant sheaf $V$?

Remarks. Hope the following is not dumb: I am pretty new to sheaf cohomology.

(a) $H^0( \cdot, \mathbb{C})$ is invariant for homotopy equivalence, so $F(X) = H^0(X, F) \simeq H^0( x, F_x) = F_x \simeq V$ via the restriction map $F(X) \to F_x $.

(b) Suppose $X$ contracts on every point $p \in X$ and let $f:X \to q$ be the map to a point. Then the canonical map $ f^{-1}f_* F \to F$ can be verified to be an isomorphism on stalks. Infact stalk maps are the restriction maps $F(X) \to F_p$, which are isomorphisms by point (a). The lefthand sheaf is constant, so we are done.

It seems to me that the problem could arise in case $X$ is not contractible on every point, which is a very counter intuitive situation.. Am I completely wrong?

Answer 1

There is a flaw in your argument :

In (a) you claim that $H^0(\cdot,\mathbb{C})$ is homotopy invariant. This is true, and this holds more generally with $\mathbb{C}$ replaced by any constant sheaf $F$, and for all degree ($H^i(\cdot,F)$ is also homotopy invariant).

In (b), you want to use (a) to prove that the sheaf $F$ is constant. But you can't : because you need $F$ to be constant in order to use (a).

The thing is (a) is false for non constant sheaves, and as a matter of fact there are non constant sheaves on every spaces (except $\emptyset$ and the one point space).

By the way, your problem does not hold as stated. You need more that $F_p\simeq V$ for all $p\in X$. For example, on $X=\mathbb{R}$ (which is contractible), $j_!V\oplus i_*V$ is a counter example (here $j:U\rightarrow X$ is the inclusion of an open subset, and $i:X\setminus U\rightarrow X$ the incluson of the closed complement). Another one : $\bigoplus_{x\in X}x_*V$. Even the sheaf of continuous function on $\mathbb{R}$ is not constant (though stalks are not finite dimensional in this case).

The correct statement is the following : on a simply-connected space (not necessarily contractible), every locally constant sheaf is constant. (Stalks are not assumed to be $\mathbb{C}$-vector spaces, nor finite dimensional)