Show that $\{x\} \times S^n \hookrightarrow S^n \times S^n$ is not contractible
It is correct to say that $\{x\} \times S^n$ is diffeomorphic to $S^n$ then it would be enough to prove that $S^n$ is contractible in $S^n \times S^n$ intuitively I can understand that a copy of $S^n$ embedded in $S^n \times S^n$ cannot be contracted as $S^n \times S^n$ has a hole in it (thinking of e.g. $S^{1} \times S^{1}$ ) so there I will have problems. I am trying to find a contradiction supposing that the embedding $j:S^n \rightarrow S^n \times S^n$ is homotopic to a constant but I have not been successful, I am trying this way because in my course we have not touched homology or homotopy groups, I think that not being able to use these tools increases the difficulty. for the moment I can only basically use the tools in the first chapter of Guillemin/Pollack's book, Sard's theorem, Whitney's embedding theorem, stability theorem, transversality, etc
Any help or suggestion I will be very grateful
