About HAP: Covering Spaces

Joint work with Kelvin Killeen

Let \(Y\) denote a finite regular CW-complex. Let \(\widetilde Y\) 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 \(C_\ast\widetilde Y\) of \(\widetilde Y\) 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 \(C_\ast \widetilde Y\) 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 \(Y\simeq S^1 \times S^1\times S^1\) 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 \(C_\ast\widetilde Y\) homotopy equivalent to the chain complex of the universal cover of \(Y\). The chain complex \(C_\ast\widetilde Y\) has just \(8\) free generators.

The next commands construct a subgroup \(H < \pi_1Y\) of index \(50\) and the chain complex \(C_\ast\widetilde Y\otimes_{\mathbb ZH}\mathbb Z\) which is homotopy equivalent to the cellular chain complex \(C_\ast\widetilde Y_H\) of the \(50\)-fold cover \(\widetilde Y_H\) of \(Y\) corresponding to \(H\).

General theory implies that the \(50\)-fold covering space \(\widetilde Y_H\) should again be homotopy equivalent to a product of three circles. As a check for this, the following commands establish that \(\widetilde Y_H\) has the same integral homology as \(S^1\times S^1\times S^1\).

Spun knots and the Satoh tube map

By 'spinning' a link \(K \subset \mathbb R^3\) about a plane \( P\subset \mathbb R^3\) with \(P\cap K=\emptyset\), we obtain a collection \(Sp(K)\subset \mathbb R^4\) of knotted tori. The following commands produce the two tori obtained by spinning the Hopf link \(K\), and then verify that these two tori have the correct homology and fundamental group. (Since the space \(Sp(K)\) is not path-connected its fundamental group is that of a single torus.). The final commands show that the space \(Y=\mathbb R^4\setminus Sp(K) = Sp(\mathbb R^3\setminus K)\) is connected with fundamental group \(\pi_1Y = \mathbb Z\times \mathbb Z\).

An alternative embedding of two tori \(L\subset \mathbb R^4 \) can be obtained by applying the 'tube map' of Shin Satoh to a 'welded Hopf link' [ Satoh, Shin. Virtual knot presentation of ribbon torus-knots. J. Knot Theory Ramifications 9 (2000), no. 4, 531--542]. The following commands construct the complement \(W=\mathbb R^4\setminus L\) of this alternative embedding and show that \(W \) has the same fundamental group and integral homology as \(Y\) above.

Despite having the same fundamental group and integral homology groups, the above two spaces \(Y\) and \(W\) were shown by Louis Kauffman and Jo\(\tilde a\)o Faria Martins to be not homotopy equivalent [Kauffman, Louis H. and Faria Martins, Jo\(\tilde a\)o. Invariants of welded virtual knots via crossed module invariants of knotted surfaces. Compos. Math. 144 (2008), no. 4, 1046--1080]. Their technique involves the fundamental crossed module derived from the first three dimensions of the universal cover of a space, and counts the representations of this fundamental crossed module into a given finite crossed module. This homotopy inequivalence is recovered by the following commands which also involve the first three dimensions of the universal cover of a space.

We mention that by tying a knot in a length of string, positioning the knotted string so that its end points lie on the x-axis and no other points lie in the xy-plane, and then rotating about the xy-plane, we obtain a knotted sphere in \(\mathbb R^4\). The following code illustrates this construction by computing invariants of the complement of the spun trefoil knot.

Cohomology with local coefficients

The \(\pi_1Y\)-equivariant cellular chain complex \(C_\ast\widetilde Y\) of the universal cover \(\widetilde Y\) of a regular CW-complex \(Y\) can be used to compute the homology \(H_n(Y,A)\) and cohomology \(H^n(Y,A)\) 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 \(\pi_1Y = \langle x,y : xyx=yxy \rangle\). 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 $$\begin{array}{lcl} H_0(Y,A) &= &\mathbb Z_2\, ,\\ H_1(Y,A) &= &\mathbb Z_3\, ,\\ H_2(Y,A) &= &\mathbb Z\, .\end{array}$$

Finite covers as regular CW-complexes

We next construct a 4-dimensional CW-complex \(Y\simeq S^1\times S^1\) homotopy equivalent to a \(2\)-dimensional torus, involving \(2304\) cells. We choose a subgroup \(H < \pi_1Y\) of index \(50\) and construct the covering space \(\widetilde Y_H\) corresponding to \(H\) as a finite regular CW-complex. The fundamental group \(\pi_1 (\widetilde Y_H)\) is shown to be free abelian on two generators. This is in keeping with the fact that \(\widetilde Y_H\) is homotopy equivalent to \(Y\).

Covering maps

It may be that we are interested in the covering map \(p:\widetilde Y_H \rightarrow Y\) and not just the covering \(\widetilde Y_H\) itself. As an illustration we construct the map \(p\) for \(Y\) homotopy equivalent to a torus, for \(H<\pi_1Y\) a subgroup with $$\pi_1Y / H \cong \mathbb Z_3 \oplus \mathbb Z_3\,,$$ and for \(p\) the corresponding covering map.

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

Distinguishing between the granny knot and reef knot

The granny knot is the sum of the trefoil knot and its mirror image. The reef knot is the sum of two identical copies of the trefoil knot. The following commands show that the degree \(1\) homology homomorphisms $$H_1(p^{-1}(B),\mathbb Z) \rightarrow H_1(\widetilde X_H,\mathbb Z)$$ distinguish between the homeomorphism types of the complements \(X\subset \mathbb R^3\) of the granny knot and the reef knot, where \(B\subset X\) is the knot boundary, and where \(p\colon \widetilde X_H \rightarrow X\) is the covering map corresponding to the finite index subgroup \(H < \pi_1X\). More precisely, \(p^{-1}(B)\) is in general a union of path components $$p^{-1}(B) = B_1 \cup B_2 \cup \cdots \cup B_t\ .$$ The function FirstHomologyCoveringCokernels(f,c) inputs an integer \(c\) and the inclusion \(f\colon B\hookrightarrow X\) of a knot boundary \(B\) into the knot complement \(X\). The function returns the ordered list of the lists of abelian invariants of cokernels $${\rm coker}(\ H_1(p^{-1}(B_i),\mathbb Z) \rightarrow H_1(\widetilde X_H,\mathbb Z)\ )$$ arising from subgroups \(H < \pi_1X\) of index \(c\). To distinguish between the granny and reef knots we use index \(c=6\).

Second homotopy groups of spaces

If \(p:\widetilde Y \rightarrow Y\) is the map from the universal cover \(\widetilde Y\) of \(Y\), then the fundamental group of \(\widetilde Y\) is trivial and the Hurewicz homomorphism \(\pi_2\widetilde Y\rightarrow H_2(\widetilde Y,\mathbb Z)\) from the second homotopy group of \(\widetilde Y\) to the second integral homology of \(\widetilde Y\) is an isomorphism. Furthermore, the map \(p\) induces an isomorphism \(\pi_2\widetilde Y \rightarrow \pi_2Y\). Thus \(H_2(U,\mathbb Z)\) 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 \(H_2(\widetilde Y,\mathbb Z) \cong \pi_2Y\). 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 \(\pi_2Y \cong \mathbb Z\).

Third homotopy groups of simply connected spaces

For any simply connected space \(U\) there is an exact sequence $$ \rightarrow \pi_4\widetilde Y \rightarrow H_4(\widetilde Y,\mathbb Z) \rightarrow H_4( K(\pi_2\widetilde Y,2), \mathbb Z ) \rightarrow \pi_3\widetilde Y \rightarrow H_4(\widetilde Y,\mathbb Z) \rightarrow 0 $$ 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 \(\widetilde Y\) its universal cover, we see that \(H_4(\widetilde Y,\mathbb Z) = H_4(\widetilde Y,\mathbb Z) = 0\) since \(\widetilde Y\) is a 2-dimensional CW-space. The exact sequence implies \(\pi_3\widetilde Y \cong H_4(K(\pi_2\widetilde Y,2), \mathbb Z )\). Furthermore, \(\pi_3\widetilde Y = \pi_3 Y\). The following commands establish that $$ \pi_3Y \cong \mathbb Z\, . $$

The following commands construct a 4-dimensional simplicial complex \(Y\) with 9 vertices and 36 4-dimensional simplices, and establish that $$ \pi_1Y=0 , \pi_2Y=\mathbb Z , H_3(Y,\mathbb Z)=0, H_4(Y,\mathbb Z)=\mathbb Z, H_4(K(\pi_2Y,2), \mathbb Z) =\mathbb Z . $$

Whitehead's sequence reduces to an exact sequence $$\mathbb Z \rightarrow \mathbb Z \rightarrow \pi_3Y \rightarrow 0$$ in which the first map is \( H_4(Y,\mathbb Z)=\mathbb Z \rightarrow H_4(K(\pi_2Y,2), \mathbb Z )=\mathbb Z \). 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 \(\pi_3Y=0\).]