Visualizing surfaces bounded by loops in perfect fundamental groups

Question

Let $M$ be a manifold of dimension 3 or higher with a perfect fundamental group. By the Hurewicz theorem, $H_1(M) = 0$ which means if we consider any non-trivial loop $l \in \pi_1(M)$ as a singular 1-chain, there must be a singular 2-chain, $\alpha$, whose boundary is $l$. The union of the images of the singular simplices of $\alpha$ will be a surface in $M$ which we can perturb to an immersed surface $S_\alpha$ with $\partial \alpha = l$. This surface cannot be a disk because then we could contract $l$ along it which would remove $l$ from $\pi_1(M)$. My question is what does $S_\alpha$ look like? Is it some sort of Mobius band?