On a $PL$ embedded 2-sphere in 4-manifold
Kazuaki Kobayashi · Hokkaido Mathematical Journal · 1975
Let M^{2n} be a simply connected differentiable manifold, and let \xi\in\pi_{n}(M^{2n}) be a given homotopy class of maps S^{n}arrow M^{2n} .It is known that if n>2 , the class \xi can be represented\backslash by a differentiable imbedding f:S^{n}arrow M^{2n} .This follows from a reasoning similar to the one used by H. Whitney to prove that every differentiable n -manifold can be differentiably imbedded in Euclidean 2n space.For n=1 , let F_{p,q} be a compact connected orientable surface of genus p with q boundary components, where q may be equal to 0. Let a_{1} , \cdots , a_{p} , b_{1} , \cdots , b_{p} , C_{1} , \cdots , C_{q-1} be standard generators for the H(F_{p.q}: Z) .Here the C_{i} correspond to consistently oriented boundary circles (one is omitted because it is homologous to the sum of the others), and the a_{i} and b_{i} are standard curves on F_{p,q} , chosen so that a_{i}\cap a_{f}=b_{i} \cap b_{f}=a_{i}\cap b_{f}=\phi if i eq j and a_{i} , b_{i} intersect nicely at one point.Then S. Suzuki [5] proved the following : SUZUKI'S THEOREM.A non zero homology class \sum_{i=1}^{p}\alpha_{\hat{v}}.a_{i}+\sum_{i=1}^{p}\beta_{i}b_{i}+\sum_{i=1}^{q-1}\gamma_{i}C_{i}