I am trying to give a brief explanation in which I make use of the concept of surgery on an $m$-manifold $M$. This is along the lines of (and taken generously from) the Wikipedia entry on Surgery; I just want to make sure that I understand the concept; I am trying to be a bit […]

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} \times S^1$ in $S^2$. =For $d=3$= Can we embed $S^{2} \times S^1$ in $S^3$? =For $d=4$= Can we […]

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of $\pi/6$, $\pi/6+2\pi/3$, $\pi/6+4\pi/3$ and then gluing three intersections. question: So what is the precise mathematical formulation of […]

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