- #1
- 7,003
- 10,423
Hi: More on Prelims:
We have a map f: S^3 -->S^3 ; S^3 is the 3-sphere , given by:
(x1,x2,x3,x4)-->(-x2,-x3,-x4,-x1).
We're asked to find its degree, and to determine if f is homotopic to the identity.
I computed that f^4 ( i.e., fofofof ) is the identity, and we have that degree is
multiplicative, so that deg(f)^4=1 , so that we can narrow the choices to degf=+/- 1.
Now, I know we can also compute the induced map on top homology, and see if f is
preserving- or reversing- orientation, but I cannot tell which it is; I am trying to
use a 4-simplex , and see if this map preserves or reverses the orientation, but
I cannot see it clearly.
Another choice I am thinking of using is that f , seen as a map from R^4 to itself,
is a linear map, so that we can calculate Det f , to see if f reverses or preserves
orientation, and then maybe argue that f restricted to the subspace S^3 (unfortunately,
S^3 is not a subspace of R^4 ) has the same effect of preserving/reversing
orientation. Any Ideas/Suggestions/Comments?
see well how to do that
We have a map f: S^3 -->S^3 ; S^3 is the 3-sphere , given by:
(x1,x2,x3,x4)-->(-x2,-x3,-x4,-x1).
We're asked to find its degree, and to determine if f is homotopic to the identity.
I computed that f^4 ( i.e., fofofof ) is the identity, and we have that degree is
multiplicative, so that deg(f)^4=1 , so that we can narrow the choices to degf=+/- 1.
Now, I know we can also compute the induced map on top homology, and see if f is
preserving- or reversing- orientation, but I cannot tell which it is; I am trying to
use a 4-simplex , and see if this map preserves or reverses the orientation, but
I cannot see it clearly.
Another choice I am thinking of using is that f , seen as a map from R^4 to itself,
is a linear map, so that we can calculate Det f , to see if f reverses or preserves
orientation, and then maybe argue that f restricted to the subspace S^3 (unfortunately,
S^3 is not a subspace of R^4 ) has the same effect of preserving/reversing
orientation. Any Ideas/Suggestions/Comments?
see well how to do that