Let \(X\) be the space obtained from a torus \(S^1 \times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle of the Möbius band to the circle \(S^1 \times \{x_0\}\) in the torus. Compute \(\pi_1(X)\), describe the universal cover of \(X\), and describe the action of \(\pi_1(X)\) on the universal cover. Do the same for the space \(Y\) obtained by attaching a Möbius band to \(\mathbb{RP}^2\) via a homeomorphism from its boundary circle to the circle in \(\mathbb{RP}^2\) formed by the 1-skeleton of the usual CW structure on \(\mathbb{RP}^2\).
Solution
Let \(M\) denote the Möbius band and let \(c\) generate \[ \pi_1(M)\cong\mathbb Z. \] The boundary circle of \(M\) represents \[ c^2\in\pi_1(M). \]
1. For the torus space \(X\),\[ \pi_1(X) \cong \left\langle a,b,c\ \middle|\ [a,b]=1,\ a=c^2\right\rangle \cong \left\langle b,c\ \middle|\ [b,c^2]=1\right\rangle. \]
Let \(a,b\) be the standard generators of \[ \pi_1(S^1\times S^1)=\langle a,b\mid[a,b]=1\rangle, \] with \(a\) represented by the attaching circle \(S^1\times\{x_0\}\). Van Kampen for the union of the torus and Möbius band identifies the boundary generator \(c^2\) with \(a\). Eliminating \(a\) gives the second presentation.
2. Equivalently,\[ \pi_1(X) \cong A*_{D}C, \]where\[ A=\langle a,b\mid[a,b]=1\rangle\cong\mathbb Z^2, \quad C=\langle c\rangle\cong\mathbb Z, \quad D=\langle a\rangle=\langle c^2\rangle\cong\mathbb Z. \]
This is exactly the van Kampen pushout from 1, and both maps from the edge group \(D\) are injective in this case.
3. Construct a simply connected space \(\widetilde X\) as follows.Its incidence graph is the Bass–Serre tree \(T\) for the amalgam \(A*_D C\).For each \(A\)-vertex take a copy of \(\mathbb R^2\), for each \(C\)-vertex take a copy of the universal cover of the Möbius band, an infinite strip \(\mathbb R\times I\), and glue each incident strip boundary line to the corresponding lift of the \(a\)-circle in the adjacent plane.
The vertices of \(T\) are the cosets \[ G/A\quad\text{and}\quad G/C, \qquad G=\pi_1(X), \] and its edges are the cosets \(G/D\), with \(gD\) joining \(gA\) to \(gC\). Since \(G=A*_D C\), the normal-form theorem for amalgamated products says that this incidence graph is a tree.
The plane is the universal cover of the torus. The infinite strip is the universal cover of the Möbius band. Each lift of the attaching circle is a line, so the edge-space gluings are homeomorphisms between boundary lines and these lifted lines.
4. The natural piecewise map\[ \widetilde X\to X \]is the universal covering map.
On every plane and strip it is the standard universal covering map of the corresponding piece. Near a point of an attaching line, a small neighborhood is obtained by gluing a half-neighborhood in the strip to a neighborhood of the corresponding line in the plane exactly as in \(X\), so the map is locally a homeomorphism there as well. Thus it is a covering map.
Every vertex space is simply connected and adjacent vertex spaces meet in a line, which is contractible. The incidence graph \(T\) is a tree. Van Kampen over finite subtrees gives trivial fundamental group for every finite union, and every loop has compact image and meets only finitely many pieces. Hence \[ \pi_1(\widetilde X)=1. \]
5. The action of \(G=\pi_1(X)\) on \(\widetilde X\) is the left-translation action on the tree of spaces.
For \(g,h\in G\), the deck transformation corresponding to \(g\) sends \[ P_{hA}\longrightarrow P_{ghA}, \qquad S_{hC}\longrightarrow S_{ghC}, \qquad L_{hD}\longrightarrow L_{ghD}, \] using the compatible deck translations inside the plane and strip pieces. This action is free, preserves all gluings, and the quotient identifies all \(A\)-pieces to the torus and all \(C\)-pieces to the Möbius band, recovering \(X\).
6. For the second space \(Y\),\[ \pi_1(Y) \cong \langle a,c\mid a^2=1,\ a=c^2\rangle \cong \langle c\mid c^4=1\rangle \cong\mathbb Z/4. \]
The usual CW structure on \(\mathbb{RP}^2\) has one \(1\)-cell \(a\) and one \(2\)-cell attached by \(a^2\). The attaching circle for the Möbius band is this \(1\)-skeleton circle, while the boundary of the Möbius band represents \(c^2\). Van Kampen therefore gives \[ a=c^2, \qquad a^2=1, \] which is equivalent to \(c^4=1\).
7. The universal cover \(\widetilde Y\) is a four-sheeted cover built from one annulus and two spheres.
Restrict the universal cover to the Möbius-band subspace. The subgroup of \[ \pi_1(M)=\langle c\rangle\cong\mathbb Z \] that maps trivially to \[ \pi_1(Y)=\mathbb Z/4 \] is \(4\mathbb Z\). The corresponding four-sheeted connected cover of the Möbius band is an annulus \(A\). Its two boundary circles each map two-to-one onto the boundary of the Möbius band.
The image of \[ \pi_1(\mathbb{RP}^2)=\langle a\mid a^2=1\rangle \] inside \(\mathbb Z/4\) is the order-two subgroup \[ \langle c^2\rangle. \] Thus the inverse image of the projective-plane piece has two connected components, each the universal double cover \[ S^2\to\mathbb{RP}^2. \] In each sphere the inverse image of the \(1\)-skeleton circle is an equator mapping two-to-one onto that circle.
Glue the two boundary circles of the annulus respectively to these two equators. The resulting space is the full four-sheeted cover \(\widetilde Y\).
8. The space \(\widetilde Y\) is simply connected, hence is the universal cover.
The annulus has fundamental group generated by either boundary circle. After gluing the first sphere along its equator, that boundary loop becomes nullhomotopic because it bounds a hemisphere in the sphere. Thus van Kampen kills the annulus fundamental group. Attaching the second sphere cannot create fundamental group. Hence \[ \pi_1(\widetilde Y)=1. \]
9. The deck group of \(\widetilde Y\to Y\) is cyclic of order four.Its generator acts on the annulus by the deck transformation induced by one unit of translation in the universal strip of the Möbius band; this interchanges the two boundary circles and therefore interchanges the two sphere components.Its square preserves each sphere and acts there by the antipodal deck transformation of \(S^2\to\mathbb{RP}^2\).
The universal-cover deck group is canonically \[ \pi_1(Y)\cong\mathbb Z/4. \] The stated action is exactly the restriction of the generator \(c\) to the lifted Möbius and projective-plane pieces. After four iterations it is the identity, while no smaller positive power fixes every point.
10. This gives the requested fundamental groups, universal covers, and fundamental-group actions for both \(X\) and \(Y\).
The torus case is <1>1–<1>5 and the projective-plane case is <1>6–<1>9.