Weak Parallelisability/Strong Orientability

Question

An orientable manifold is defined by looking at small patches, looking at the interaction of local homology groups on these small patches, and focusing on when this gives us a coherent global story. If the manifold is differentiable, we can define parallelisability, choosing a basis for the tangent space which changes smoothly, which obviously implies orientability, but is stronger for differentiable manifolds. Here are some in-between conditions: $n$-manifold $M$ admits a continuous map $f:M\times S^{n-1}\to M$ such that, at all $p\in M$, $f[\{p\}\times S^{n-1}]$ corresponds to a generator of the local homology group at $p$ (as its boundary). $M$ admits a continuous map $f:M\times B^n\to M$ such that $f:(p,0)\mapsto p$, and, at all $p\in M$, $f[\{p\}\times B^n]$ corresponds to a generator of the local homology group at $p$, with $(p,q)=p$ implying $q=0$. Are these just restatements of orientability?