Articles of projective space

integral cohomology ring of real projective space

What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher’s book and Proving that the cohomology ring of $\mathbb{R}P^n$ is isomorphic to $\mathbb{Z}_{2}[x]/(x)^{n+1}$. For the graded module structure, it is obtained from the homology struture.

Type and number of points at infinity on Riemann surface

Consider a polynomial $P(z) = z^4 \in \mathbb{C}[z]$. Set-theoretically $P(z)$ has one root equal to zero. From algebraic point of view it has four roots: root zero has multiplicity four. Also we can’t draw a curve in $\mathbb{C}$ around one of such roots but not around the others. Now consider a Riemann surface $X$ given […]

Intersection of maximal subgroups isomorphic to $PGL_2(p)$ is of form $PGL_2(p_0)$

I want prove (or disprove) that given a finite group G, any two maximal subgroups that are isomorphic to $PGL(2,p)$ where p is prime, then their intersection is isomorphic $PGL(2,q_0)$ (for some $q_0$). I am aware of the fact that intersection of subfields is also a subfield. However, the first statement is unclear to me. […]

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is topologically equivalent to the circle. It is equivalent to the upper half of the circle where the two end points are glued together – i.e. another […]

“Intrinsic” treatment of projective spaces

The question I would like to ask is the following one. Consider a projective space just as a smooth manifold, e.g. $\mathbb{C}P^1$ is $S^2$. Then most maps from $S^2$ to $S^2$ even if smooth, injective and surjective do not respect the “projective space structure” (which I am not sure about how to define) and therefore […]

Vector Field in a complex projective space

This question is motivated by this answer here. Let $\mathbb{C}P^{n}$ be a complex projective space. Let $X\in\Gamma(T\mathbb{C}P^{n})$, be a vector field. It seems, by the answer I got in the mentioned link, that the zeros of $X$ are the points $[z]\in\mathbb{C}P^{n}$ such that $X([z])=\lambda[z]$, for some $\lambda$. Can anyone please make this clear for me […]

Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of $\mathbb{R}$. What about if we change the topology by consideration of $p-$adic topology on rational numbers?

rational functions on projective n space

How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree defined/regular on whole of projective space.

Cup Product Structure on the Projective Space

I am reading about cup products and am stuck on this exercise in Hatcher (3.2.5). Taking as given that $H^*(\mathbb{R}P^\infty,\mathbb{Z}_2)\simeq\mathbb{Z}_2[\alpha]$, how does one show $H^*(\mathbb{R}P^\infty,\mathbb{Z}_4)\simeq \mathbb{Z}_4[\alpha,\beta]/(2\alpha,2\beta,\alpha^2)$ with $\deg(\alpha)=1,\deg(\beta)=2$? The cohomology groups are $\mathbb{Z}_4$ in position 0 and $\mathbb{Z}_2$ in every other position. I want to find the cup product structure. To do that, I need […]

Unique line in $\mathbb{P}^3$ through a point $p$ and intersecting two disjoint lines

I’m a bit stuck with this exercise from a script I’m reading, and I’m not very familiar with projective $n$-space yet. The problem: Let $L_1$ and $L_2$ be two disjoint lines in $\mathbb{P}^3$, and let $p\in\mathbb{P}^3\smallsetminus(L_1\cup L_2)$. Show that there is a unique line $L\subseteq\mathbb{P}^3$ meeting $L_1$, $L_2$, and $p$ (i.e. such that $p\in L$ […]