I am working on Topological Complexity and I need to use the cohomology ring and the cohomology groups over some fields ($\mathbb{Q}$ and $\mathbb{Z}_2$ to be precise) of some spaces to obtain lower bounds. I don't need to compute them, just to cite books where both the cohomology rings and the cohomology groups are stated (it would be even better if they are proved or obtained). The spaces whose cohomology ring and cohomology groups I need are:

  • The sphere $\mathbb{S}^n$
  • The wedge sum of $k$ circles
  • The projective spaces (Real, Complex and Quaternionic) (The cohomology ring over $\mathbb{Z}_2$ in the real case and over the integers in the complex case are done in Hatcher Theorem 3.19 but I still need the cohomology groups). And I also need the Cohomoly ring and groups of the real case when we choose $\mathbb{Q}$ as the commutative ring, in fact, field.
  • The compact orientable surfaces of genus $g \geq 2$
  • The n-dimensional torus $T^n$

Maybe cohomoly ring could be calculated using cohomology groups but I'm starting to study cohomology so I don't know how and I can not waste pages of the dissertation on that. Moreover I am allowed to assume the cohomoly groups and cohomoly rings without proof.

So to sum up, I need references of books where both the Cohomology groups and Cohomology ring (over $\mathbb{Q}$ and $\mathbb{Z}_2$) of the spaces above are, at least, stated. If you know more than one please don't hesitate to mention more.

Thanks in advance.

  • 1
    $\begingroup$ Try the following books authors: Hatcher, Spanier, Switzer, Vick, ton Dieck, May. Note that the cohomology ring of the Torus is a tensor product of that of the 1-sphere (and this is easy to obtain), that the projective space has cohomology ring the truncated polynomials in appropriate generating degree, and the sphere has cohomology ring the dual numbers in an appropriate generating degree. $\endgroup$ – Pedro Tamaroff Mar 21 '16 at 18:57
  • $\begingroup$ The groups are just the rings but forgetting the multiplication... $\endgroup$ – Thomas Rot Mar 22 '16 at 13:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.