= Surgery on a smooth manifold
{title2=$S^{\lambda-1}\times D^{m+1-\lambda}\rightsquigarrow D^\lambda\times S^{m-\lambda}$}
A framed embedding of the left-hand region in an $m$-dimensional <manifold> specifies an index-$\lambda$ surgery. Remove its interior and glue the right-hand region along their common boundary $S^{\lambda-1}\times S^{m-\lambda}$. The <cobordism> realizing this change attaches an index-$\lambda$ <handle> of dimension $m+1$. On an oriented <surface>, index-two surgery cuts out an annular neighbourhood of a simple closed curve and caps its two boundary circles.
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