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    Homology of $S^2$ with equatorial antipodal identification, and $S^3$ with equatorial $S^2$ antipodal identification

    E-HAT-2.2-10

    Area
    Topology
    Topics
    CW Complexes, Homology
    Status
    draft

    Let \(X\) be the quotient space of \(S^2\) under the identifications \(x \sim -x\) for \(x\) in the equator \(S^1\). Compute the homology groups \(H_i(X)\). Do the same for \(S^3\) with antipodal points of the equatorial \(S^2 \subset S^3\) identified.

    Solution

    Let \(X_2\) be the quotient of \(S^2\) obtained by identifying antipodal points on the equator.

    1. The space \(X_2\) has a CW structure with one \(0\)-cell, one \(1\)-cell, and two \(2\)-cells, with both \(2\)-cells attached to the \(1\)-skeleton \(\mathbb{RP}^1\cong S^1\) by maps of degree \(2\) up to sign.

    The quotient equator is \(S^1/(x\sim-x)=\mathbb{RP}^1\cong S^1\). The northern and southern open hemispheres descend to two open \(2\)-cells. On the boundary of either hemisphere the quotient map \(S^1\to\mathbb{RP}^1\) is the standard double covering, hence has degree \(\pm2\) according to orientation choices.

    Thus the cellular chain complex is \[ 0\longrightarrow\mathbb Z^2 \xrightarrow{d_2}\mathbb Z \xrightarrow{0}\mathbb Z\longrightarrow0, \qquad d_2(a,b)=2a\pm2b. \] Therefore \[ \boxed{ H_i(X_2)\cong \begin{cases} \mathbb Z,&i=0,2,\\ \mathbb Z_2,&i=1,\\ 0,&\text{otherwise}. \end{cases}} \] Indeed, \(\operatorname{im}d_2=2\mathbb Z\) and \(\ker d_2\cong\mathbb Z\).

    Now let \(X_3\) be the quotient of \(S^3\) obtained by identifying antipodal points on the equatorial \(S^2\).

    2. The \(2\)-skeleton of \(X_3\) is \(\mathbb{RP}^2\), and the two hemispheres give two \(3\)-cells attached by the standard covering map\[ S^2\longrightarrow\mathbb{RP}^2. \]

    The equator quotient is exactly \(\mathbb{RP}^2\). Each open hemisphere of \(S^3\) is a \(3\)-ball, and its boundary is identified with \(\mathbb{RP}^2\) by the antipodal quotient map.

    3. The cellular boundary \(d_3:\mathbb Z^2\to\mathbb Z\) is zero.

    For the standard CW structure on \(\mathbb{RP}^3\), the attaching map of the \(3\)-cell is the same quotient map \(S^2\to\mathbb{RP}^2\), and its cellular boundary coefficient is \[ 1+(-1)^3=0. \] Equivalently, after collapsing \(\mathbb{RP}^1\), the attaching map \(S^2\to S^2\) has degree zero. Each of the two \(3\)-cells in \(X_3\) has this attaching map, so both columns of \(d_3\) vanish.

    The cellular chain complex is therefore \[ 0\longrightarrow\mathbb Z^2 \xrightarrow{0}\mathbb Z \xrightarrow{2}\mathbb Z \xrightarrow{0}\mathbb Z\longrightarrow0. \] Hence \[ \boxed{ H_i(X_3)\cong \begin{cases} \mathbb Z,&i=0,\\ \mathbb Z_2,&i=1,\\ \mathbb Z^2,&i=3,\\ 0,&\text{otherwise}. \end{cases}} \]

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