This is an exercise 6.43 from Bott's Topology book.I'm not asking it's solution,but ask something relative to it.(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex] \pi :E \to M\ [/tex] be an oriented rank 2 bundle.[tex] \Phi = d(\rho (r)\frac{{d{\theta _\alpha }}}{{2\pi }}) + \frac{1}{{2\pi i}}d(\rho (r){\pi ^*}\sum\limits_\gamma {{\rho _\gamma }d\log {g_{\gamma \alpha }})} \ [/tex] is the explicit formula for the Thom class.As we know wedging with the Thom class is an isomorphism [tex] () \wedge \Phi :{H^*}(M) \to H_{cv}^{* + 2}(E)\ [/tex] .The problem is to find the class u on M such that

[tex] {\Phi ^2} = \Phi \wedge {\pi ^*}u{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} in{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} H_{cv}^*(E){\kern 1pt} [/tex]

My confusion is: What does [tex] {\Phi ^2}\ [/tex]means?Does it mean [tex] {\phi ^2}=\phi \wedge \phi [/tex]? If it is yes,then this is my caculation [tex] \Phi = d\rho (r)\frac{{d{\theta _\alpha }}}{{2\pi }} + \frac{1}{{2\pi i}}d\rho (r){\pi ^*}\sum\limits_\gamma {{\rho _\gamma }d\log {g_{\gamma \alpha }}} + \frac{1}{{2\pi i}}\rho (r){\pi ^*}\sum\limits_\gamma {d{\rho _\gamma }d\log {g_{\gamma \alpha }}} \ [/tex] and [tex] {\Phi ^2} = 2d\rho (r)\frac{{d{\theta _\alpha }}}{{2\pi }}\frac{1}{{2\pi i}}\rho (r)({\pi ^*}\sum\limits_\gamma {d{\rho _\gamma }d\log {g_{\gamma \alpha }})} + 2\frac{1}{{2\pi i}}d\rho (r)({\pi ^*}\sum\limits_\gamma {{\rho _\gamma }d\log {g_{\gamma \alpha }}} )\frac{1}{{2\pi i}}\rho (r)({\pi ^*}\sum\limits_\gamma {d{\rho _\gamma }d\log {g_{\gamma \alpha }})} + {(\frac{1}{{2\pi i}}\rho (r){\pi ^*}\sum\limits_\gamma {d{\rho _\gamma }d\log {g_{\gamma \alpha }}} )^2}\ [/tex],did I do it right?How can I find the class u?Could somebody give me hints?

Thank you!

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How to find the class in H^*(M)?

Loading...

Similar Threads - find class | Date |
---|---|

A Can I find a smooth vector field on the patches of a torus? | Oct 9, 2017 |

A Why the Chern numbers (integral of Chern class) are integers? | Oct 29, 2016 |

Find the point on a line that is X distance away from a plane | Jun 1, 2014 |

Find the point on a cone that is farthest in a given direction | May 20, 2014 |

Lagrangian Multipliers to find maximum and minimum values | Apr 9, 2014 |

**Physics Forums - The Fusion of Science and Community**