## When is the quotient of a manifold by a discrete group of diffeomorphisms a diffeological covering space?

I was reading An Introduction to Diffeology by Patrick Iglesias-Zemmour and he defines a diffeological covering space as a diffeological fiber bundle with discrete fiber. My question: Consider a manifold $M$ and a discrete subgroup $G$ of $\mathrm{Diff}(M)$ acting on $M$ freely. When is the quotient morphism $M \to M/G$ a diffeological covering space? I … Read more

## Descent theory, fibrations, and bundles

In the very last page of Janelidze and Tholen’s paper Beyond Barr Exactness: Effective Descent Morphisms, the authors relate the theory of fiber bundles (and covering spaces in particular) to descent and the lifting properties defining fibrations (in topology). Below is an excerpt from the final two pages of the paper. I don’t understand section … Read more

## Cellular homology of the universal cover

Let X be a connected pointed CW complex. Let ˜X be its universal covering space and G=π1(X). Lets denote (CCell∗(˜X),d) the cellular chain complex associated to ˜X. By construction each CCelln(˜X) is a free (left) G-module and the differential operators d:CCelln+1(˜X)→CCelln(˜X) are (left) G-equivariant. Question: Since each CCelln(˜X) is a free (left) G-module, there is … Read more

## Path-lifting property: function space interpretation

I asked this question on math.SE, but even with a bounty, there were no answers/comments. I hope this is not too low-level for this site. Suppose I have a covering map π:E→B, and a path in B, which is just a map f from I=[0,1] to B; then I know I can lift this path … Read more

## The Classification of all spaces for which XX is a covering space

A well-known problem is to classify all covering spaces of a topological space X. For example, if X is a semi-locally simply connected space, then each equivalent class of a covering space of X is corresponding to conjugacy class of a subgroup of π1(X). Now my question is that: Is there any classification of all … Read more

## To what extent are geometric methods being used to attack the inverse Galois problem?

My limited knowledge so far is that some groups have been constructed geometrically using the theory of covering spaces, then applying Hilbert irreducibility. Is there a deeper way in which inverse Galois theory is connected to, for example, Grothendieck’s algebraic geometry and the study of etale fundamental groups? Answer AttributionSource : Link , Question Author … Read more

## Lifting local compactness to a covering space

(I decided to repost this from MathSE, since the question seems to not be as easy as I had thought) NB: In this question, local compactness is used in its weak form, i.e. in a locally compact space, every point has a compact neighbourhood. Also, T3 is the weaker condition of the pair T3/regular. Let … Read more

## Construction of the universal covering space of the etale homotopy type Et(X)Et(X)

Let X be a nice scheme (additional assumptions could be added), and let Et(X) be its (Artin-Mazur) etale homotopy type. I am looking for a/the scheme Y over X whose etale homotopy type Et(Y) will be the topological universal cover of Et(X). By definition Et(X) is the geometric realization of a simplicial set, and it … Read more

## Nondegeneracy of kernel of map on homology induced by covering of surfaces

Let f:X→Y be a finite covering map between compact oriented surfaces and let K be the kernel of the induced map f∗:H1(X)→H1(Y). Here homology has rational coefficients. Question: must K be nondegenerate with respect to the symplectic algebraic intersection pairing ω on H1(X)? In other words, is it true for all nonzero k∈K, there exists … Read more

## Sufficient condition for coverings between non-orientable surfaces

Let Xk be the connected sum of k projective planes. I am interested in necessary and sufficient conditions for the existence of a covering π:Xk′→Xk, where k and k′ are integers. A necessary condition is that the Euler characteristic of Xk′ is a multiple of the Euler characteristic of Xk. Though obtaining a sufficient condition … Read more