Timeline for Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 3, 2016 at 12:06 | history | edited | Neil Strickland | CC BY-SA 3.0 |
added 2042 characters in body
|
| Apr 30, 2016 at 15:11 | comment | added | Neil Strickland | OK, I see your point. I might look at this again tomorrow. | |
| Apr 30, 2016 at 14:37 | comment | added | Tyrone | The fibration resulting from applying the functor $Map_*(-,X)$ to a cofiber sequence only works in the pointed topological category: it needs all spaces to be based. Thus the fibre it produces is really the fibre over the basepoint component, in this case $Map_*^{(0,0)}(\mathbb{C}P^2,BU(2))$ containing the trivial map. There is no particular reason why the fibre over the non-basepoint component $Map_*^{(1,0)}(\mathbb{C}P^2,BU(2))$ should be of the same homotopy type. In fact they are not and $Map_*^{(1,0)}(\mathbb{C}P^2,BU(2))\not\simeq Map_*^{(0,0)}(\mathbb{C}P^2,BU(2))$ (calculate $\pi_4$). | |
| Apr 30, 2016 at 14:35 | comment | added | Tyrone | The problem with this is that the map induced by $BS^3\rightarrow BU(2)$ takes the $l$ component of $Map_*(\mathbb{C}P^2,BS^3)$ into the $(0,l)$ component of $Map_*(\mathbb{C}P^2,BU(2))$, i.e. $Map_*^l(\mathbb{C}P^2,BS^3)\rightarrow Map_*^{(0,l)}(\mathbb{C}P^2,BU(2))$. The coaction induced by the pinch map $\mathbb{C}P^2\rightarrow S^4$ then may be used to get homotopy equivalences $Map_*^{(0,l)}(\mathbb{C}P^2,BU(2))\simeq Map_*^{(0,0)}(\mathbb{C}P^2,BU(2))$ And these later spaces certainly sit in the fiber sequence you defined but consider the $(1,l)$ component in which I am interested. | |
| Apr 29, 2016 at 17:57 | history | answered | Neil Strickland | CC BY-SA 3.0 |