Intereting Posts

Proving that $\gcd(ac,bc)=|c|\gcd(a,b)$
Why generalize vector calculus with $k$-forms instead of $k$-vectors?
What's a good book on advanced linear algebra?
Proving ${p-1 \choose k}\equiv (-1)^{k}\pmod{p}: p \in \mathbb{P}$
composition of $L^{p}$ functions
Why is $L_A$ not $\mathbb K$ linear (I can prove that it is)
Indefinite Integral of Floor Function Integration by Substitution
Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
Right and left inverse
Combinatorial Interpretation of a Certain Product of Factorials
What is a good book for a second “course” in group theory?
Sum of reciprocal prime numbers
Can we found mathematics without evaluation or membership?
How many words of length $n$ can we make from $0, 1, 2$ if $2$'s cannot be consecutive?
Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares

It is known that if $X$ is a finite CW complex and if $Y \to X$ is a $n$-sheeted covering then $Y$ is a finite CW complex and $\chi(Y)=n \cdot \chi(X)$.

More generally, Euler characteristic can be defined as $\chi(X)= \sum_i(-1)^i \text{rank}(H_i(X))$ when all but finitely many homology groups are trivial (ie. when $X$ is of bounded finite type). In this case, does the preceding statement still hold?

If $X$ is of bounded finite type and if $Y \to X$ is a $n$-sheeted covering, is it true that $Y$ is of bounded finite type and $\chi(Y)=n \cdot \chi(X)$?

A weaker problem is: Is there a homological proof for CW complexes?

- Do trivial homology groups imply contractibility of a compact polyhedron?
- What are necessary and sufficient conditions for the product of spheres to be paralellizable?
- The Euler characteristic & a cube with holes?
- How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?
- Why is the cohomology of a $K(G,1)$ group cohomology?
- Topology on the space of paths
- CW complexes and manifolds
- Strong deformation retract
- Understanding the trivialisation of a normal bundle
- Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

From the Leray-Serre spectral sequence for a covering map $Y\to X$, which is a fibration with discrete fibers, we get an isomorphism $H^p(Y)\cong H^p(X,\mathcal H^0)$, where $\mathcal H^0$ denotes the local system of coefficients which at each point of $X$ has group equal to $H^0(p^{-1}(x))$.

If the covering is of $n$ sheets, then $\mathcal H^0$ is locally $\mathbb Z^n$, with the fundamental group of $X$ acting by permutation of the standard basis according to the monodromy permutation representation.

Now the local system $\mathcal H^0$ corresponds to a sheaf $\mathcal F$ on $X$, and for sensible $X$ (paracompact, say), one can compute singular cohomology with coefficients in the local system as sheaf cohomology with coefficients on the sheaf $\mathcal F$. If $X$ has a good finite cover $\mathcal U$ (in the sense of the book of Bott-Tu) then one can compute sheaf cohomoogy $H^p(X,\mathcal H^0)$ as the Cech cohomology $H^p(\mathcal U,\mathcal H^0)$. Looking at the complex which computes this by definition, we see that the Euler characteristic of $H^p(\mathcal U,\mathcal H^0)$, and therefore of $H^\bullet(X,\mathcal H^0)$, is $n$ times that of $H^\bullet(X,\mathbb Z)$. Notice that the existence of good covers implies being of bounded finite type, as you say (but I think it even implies that the space of of the homotopy type of a CW-complex, namely the nerve of the good covering… so all this might not get us much)

The fact that the Euler characteristic of a sensible space with coefficients on a local system of coefficients which locally looks like $\mathbb Z^n$ is $n$ times that of the space should be written down somewhere, but I cannot find it now.

There is this answer by Matt but he does not give a reference.

- Logic Puzzle of the age of three sons
- Find three real orthogonal matrices of order $3$ having all integer entries.
- Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
- Why does the natural ring homomorphism induce a surjective group homomorphism of units?
- Calculate the degree of the extension $$
- Proving by induction that $1^3 + 2^3 + 3^3 + \ldots + n^3 = \left^2$
- A generalization of Kirkman's schoolgirl problem
- Finding the roots of an octic
- Integrals involving reciprocal square root of a quartic
- The Matrix Equation $X^{2}=C$
- Is there “essentially only 1” Jordan arc in the plane?
- What is wrong with this fake proof $e^i = 1$?
- Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.
- A uniformly continuous function between totally bounded uniform spaces
- A stronger version of discrete “Liouville's theorem”