Concerning the Spanier group relative to an open cover

Question

Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$‎. Spanier defined $\pi (\mathcal{U}‎, ‎x)$ to be the subgroup of $\pi_1 (X‎, ‎x)$ which contains all homotopy classes having representatives of the following type‎: $ ‎\prod_{j=1}^{n}u_j *v_j * u^{-1}_{j}‎, ‎$ ‎where $u_j$'s are paths (starting at the base point $x$) and each $v_j$ is a loop inside one of the neighbourhoods $U_i \in \mathcal{U}$‎.

‎If an open cover $‎\mathcal{U}$ is a refinement of an open cover $‎\mathcal{V}$‎, then $\pi (‎\mathcal{U}‎, ‎x) \subset \pi (‎\mathcal{V}‎, ‎x)$‎.

My question is that:

If $[f][g]\in \pi (‎\mathcal{U}‎, ‎x)$ for $[f],[g]\in \pi_1 (X,x)$, then is there any refinement $\mathcal{V}$ of $\mathcal{U}$ so that $[f]\in \pi (\mathcal{V},x)$?

Answer 1

No. Let $X_1=X_2=X_3=S^1$ be copies of the unit circle and consider $X_1\vee X_2\vee X_3$ with wedge basepoint $x$. Let $\gamma_i$ be a loop traversing $X_i$ whose homotopy class generates $\pi_1(X_i,x)$. Construct $X$ by attaching a 2-cell to $X_1\vee X_2\vee X_3$ by the attaching loop $\gamma_1\ast\gamma_2\ast \gamma_{3}^{-1}$.

Consider an open cover $\mathscr{U}$ of $X$ consisting of

• the interior of the attached 2-cell,
• an open neighborhood $U$ of $X_3$ that deformation retracts onto $X_3$ (i.e. $X_3$ with a partial collar),
• Small simply connected sets in $X$ whose union contains $X_1\vee X_2$

Notice that $U$ is the only non-simply connected set in $\mathscr{U}$ and so $\pi_1(\mathscr{U},x)$ is the normal subgroup of $\pi_1(X,x)$ generated by $[\gamma_3]$. In particular, $[\gamma_1][\gamma_2]=[\gamma_3]\in \pi_1(\mathscr{U},x)$ but $[\gamma_1]\notin \pi_1(\mathscr{U},x)$. It follows that if $\mathscr{V}$ is a refinement of $\mathscr{U}$, then $[\gamma_1]$ can't lie in $\pi_1(\mathscr{V},x)$ either since it's a subgroup of $\pi_1(\mathscr{U},x)$.