Let $H_1(X,\mathbb{Z})$ the first homology group of $X$ with integral coefficients, where $X$ is a topological space. In particular I consider $X=\mathbb{S}^1 \vee \mathbb{RP}^2$. Let $x_0$ be the intersection point between $\mathbb{S}^1$ and $ \mathbb{RP}^2$.
I have three questions I'm trying to answer:
1) Compute $\pi_1(X,x_0)$.
2)Compute $H_1(X,\mathbb{Z})$ where it is well known $H_1(X,\mathbb{Z})=\pi_1(X,x_0)/\bigl[\pi_1(X,x_0),\pi_1(X,x_0) \bigr]$ ($\bigl[\pi_1(X,x_0),\pi_1(X,x_0) \bigr]$ is the commutator subgroup).
3) Find a topological space $Y$ and a covering space $(\tilde Y,p) \,$ $p\colon \tilde Y \to Y$ such that Aut$_Y(\tilde Y)$ (deck transformation group ) is isomorphic to $H_1(X,\mathbb{Z})$.
1) Using Van Kampen's theorem it's easy to check that $\pi_1(X,x_0)= \mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$.
2) I have no idea how to compute this (Intuitively I suppose it could be $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z}$)
3)If i knew $H_1(X,\mathbb{Z})$ I could find easily such a space $Y$
Any ideas?
P.S. At University we studied $H_1(X,\mathbb{Z})$ roughly and in particolar I know only the definition (written above) and I know nothing about rank, torsion, group representations.
Best Answer
Use the theorem that for a path-connected space, $H_1(X,\Bbb Z)$ is the Abelianisation of $\pi_1(X,x_0)$. From your calculation of $\pi_1$ you immediately get that $H_1$ is $\Bbb Z\times(\Bbb Z/2\Bbb Z)$.