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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

5 votes

Characterization of a non-trivial non-peripheral element of the free homotopy classes of a c...

I don't think so. Let $\Sigma$ be the pair of pants, and $\alpha$ the curve, both pictured below. Then $x=[\alpha]$ is non-trivial since it has non-zero winding number with one of the holes. It is non …
Johannes Huisman's user avatar
5 votes
Accepted

Inequivalent free $\Bbb Z/n\Bbb Z$-actions on orientable compact bordered surface

Did you try to use the double $T$ of the surface $S$? Any fixed point-free action of $\mathbf Z/n$ on $S$ induces a fixed point-free action on the closed orientable surface $T$. Moreover, the induced …
Johannes Huisman's user avatar
12 votes

Quotient of solid torus by swapping coordinates on boundary

I believe that your quotient space can be seen as the quotient of the $3$-sphere $S^3$ in $\mathbf C^2$ by the action of complex conjugation. The $3$-sphere $S^3$ can be identified with the join $S^1\ …
Johannes Huisman's user avatar