About HAP: Covering Spaces

Joint work with Kelvin Killeen

Let $Y$ denote a finite regular CW-complex. Let $U$ denote its universal covering space. The covering space inherits a regular CW-structure which can be computed and stored using the data type of a $\pi$₁Y-equivariant CW-complex. The cellular chain complex $CU$ of $U$ can be computed and stored as an equivariant chain complex. Given an admissible discrete vector field on $Y$, we can endow $Y$ with a smaller non-regular CW-structre whose cells correspond to the critical cells in the vector field. This smaller CW-structure leads to a more efficient chain complex $CU$ involving one free generator for each critical cell in the vector field.

Cellular chains on the universal cover

The following commands construct a $6$-dimensional regular CW-complex $Y\backsimeq S1\times \; S1\times S1$ homotopy equivalent to a product of three circles.

The CW-somplex $Y$ has $110592$ cells. The next commands construct a free $\pi$₁Y-equivariant chain complex $CU$ homotopy equivalent to the chain complex of the universal cover of $Y$. The chain complex $CU$ has just $8$ free generators.

The next commands construct a subgroup ₁Y of index $50$ and the chain complex $CU\otimes$_ℤHℤ which is homotopy equivalent to the cellular chain complex $CU$_H of the $50$-fold cover $U$_H of $Y$ corresponding to $H$.

General theory implies that the $50$-fold covering space $U$_H should again be homotopy equivalent to a product of three circles. As a check for this, the following commands establish that $U$_H has the same integral homology as $S1\times \; S1\times S1$

Cohomology with local coefficients

The $\pi$₁Y-equivariant cellular chain complex $CU$ of the universal cover $U$ of a regular CW-complex $Y$ can be used to compute the homology $H$_n(Y,A) and cohomology $Hn(Y,A)$ of $Y$ with local coefficients in a $\mathbb{Z}\pi$₁Y-module $A$. To illustrate this we consister the space $Y$ arising as the complement of the trefoil knot, with fundamental group $\pi$₁Y = <x,y : xyx=yxy >. We take $A=\; \mathbb{Z}$ to be the integers with non-trivial $\pi$₁Y-action given by $x.1=-1,\; y.1=-1$. We then compute

Finite covers as regular CW-complexes

We next construct a 4-dimensional CW-complex ₁Y of index $50$ and construct the covering space $U$_H corresponding to $H$ as a finite regular CW-complex. The fundamental group $\pi$₁(U_H) is shown to be free abelian on two generators. This is in keeping with the fact that $U$_H is homotopy equivalent to $Y$.

Covering maps

It may be that we are interested in the covering map $p:U$_H → Y and not just the covering $U$_H itself. As an illustration we construct the map $p$ for $Y$ homotopy equivalent to a torus, for ₁Y a subgroup with

and for $p$ the corresponding covering map.

The covering map $p$ induces homomorphisms $H$_n(p):H_n(W,Z) → H_n(Y,Z) on integral homology. These homomorphisms, together with their cokernels, can be computed as follows.

Second homotopy groups of spaces

If $p:U\; \to \; Y$ is the map from the universal cover $U$ of $Y$, then the fundamental group of $U$ is trivial and the Hurewicz homomorphism $\pi$₂U-->H₂(U,ℤ) from the second homotopy group of $U$ to the second integral homology of $U$ is an isomorphism. Furthermore, the map $p$ induces an isomorphism $\pi$₂U → π₂Y. Thus $H$₂(U,ℤ) is isomorphic to the second homotopy group $\pi$₂Y.

If the fundamental group of $Y$ happens to be finite, then in principle we can calculate $H$₂(U.ℤ) ≅ π₂Y. We illustrate this computation for $Y$ equal to the real projective plane.

The above computation shows that the space $Y$ has infinite cyclic second homotopy group $\pi$₂Y = ℤ .

Third homotopy groups of simply connected spaces

For any simply connected space $U$ there is an exact sequence

due to J.H.C.Whitehead. Here $K(\pi$₂U,2) is an Eilenberg-MacLane space with second homotopy group equal to $\pi$₂U.

Continuing with the above example where $Y$ is the real projective plane and $U$ its universal cover, we see that $H$₄(U,ℤ) = H₄(U,ℤ) = 0 since $U$ is a 2-dimensional CW-space. The exact sequence implies $\pi$₃U ≅ H₄(K(π₂U,2), ℤ ). Furthermore, $\pi$₃U = π₃Y since $U$ is the universal cover. The following commands establish that

The following commands construct a 4-dimensional simplicial complex $Y$ with 9 vertices and 36 4-dimensional simplices, and establish that

Whitehead's sequence reduces to an exact sequence

in which the first map is $H$₄(Y,ℤ)=ℤ → H₄(K(π₂Y,2), ℤ)=ℤ . In order to determine $\pi$₃Y it remains compute this first map. This computation is currently not available in HAP.

[The simplicial complex in this second example is due to W. Kiihnel and T. F. Banchoff and is of the homotopy type of the complex projective plane. So, assuming this extra knowledge, we have $\pi$₃Y=0.]