# Concerning the Spanier group relative to an open cover

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 $$[f]\in \pi (\mathcal{V},x)$$?

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 $$\pi_1(\mathscr{U},x)$$.