3
votes
1answer
63 views

Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
3
votes
0answers
54 views
+100

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
2
votes
0answers
33 views

Surjectivity and exactness on higher homotopy groups

If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$. If $f\colon X \rightarrow Y$ is ...
1
vote
0answers
33 views

Real Spectrum of a Commutative Ring

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance. $\text{Spec}_{r}(A)$
3
votes
1answer
36 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
4
votes
1answer
52 views

Working out an example of a Chern class

I'm trying to understand page 161 of Fulton's "Young tableaux" in an explicit example. I'm looking at flags in $\mathbb{C}^4$, which I think of as flags in $\mathbb{CP}^3$ (and I'm really just able ...
0
votes
0answers
39 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
3
votes
0answers
30 views

Why is this Milnor fiber homeomorphic to a cylinder?

Let $f:(\mathbb C^2,0)\to (\mathbb C,0)$ be a holomorphic function with a critical point at the origin. Let us denote by $X_0$ the fiber $f^{-1}(0)$, in which, as we said, the point $0\in X_0$ is a ...
2
votes
1answer
25 views

Density, Irreducible Topological Space

Let X is a irreducible noetherian topological space, and $U \subseteq X$ is a nonempty open subset. If $B \subseteq U$ is dense, then so is $B \subseteq X$. Is this true or false, and why? Here ...
0
votes
0answers
11 views

Orthogonal Array Planes

In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ...
0
votes
1answer
32 views

Correspondence of Grassmannian cells

I have shown that the correspondence $f: X \rightarrow \mathbb{R} \oplus X$ defines an embedding of the Grassmannian $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ into $\mathrm{Gr}_{k+1}(\mathbb{R} \oplus ...
3
votes
1answer
37 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
6
votes
1answer
56 views

Relation between different definitions of degree in complex geometry

Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
1
vote
1answer
51 views

Computing the Todd class of projective space.

As an exercise I'm trying to verify that for $X=\Bbb{P}_k^n$, where $k$ is an algebraically closed field, we have $$\operatorname{td}(X)=\left(\frac{\epsilon}{1-e^{-\epsilon}}\right)^{n+1},$$ where ...
2
votes
0answers
58 views

Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
3
votes
1answer
68 views

Why are there cubics in a Calabi-Yau manifold?

I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.) ...
0
votes
1answer
70 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
2
votes
1answer
89 views

Is there a “geometric” interpretation of inert primes?

I am currently learning about étale morphisms and how they behave a lot like covering maps. Now if I'm in the example of the ring of integers of a number field, it is possible to think of it as a ...
6
votes
3answers
128 views

Are the fibers of a flat map homotopy equivalent?

At the end of the Wikipedia article on Deformation Retract, there is the following sentence: Two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger ...
2
votes
0answers
28 views

Cech cohomology [duplicate]

There are 2 complexes computing Cech cohomology. The difference between them is that in the second one we require skew symmetry when you change the order of indices. How to show that they are ...
5
votes
0answers
51 views

Serre Fibration long exact sequence

I want to know if there exists something like the long exact sequence on the following case: Let $p : E\rightarrow B$ a continuous surjective map such that there exists an open dense subset $U$ of ...
0
votes
0answers
60 views

Fundametal Group and Étale Fundamental Group

For what kind of schemes the fundamental group and the étale fundamental group coincide? Is there a relation between this groups on a variety? I'm interested on toric varieties (more generally, ...
4
votes
1answer
72 views

Does the topological fundamental group on schemes provide any interesting information?

I'm just learning my first things about the étale fundamental group. And while I see that the usual fundamental group is not really a "natural" thing to do in the category of schemes I wonder wether ...
7
votes
1answer
69 views

Homology of stack points

This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the classifying stack $\mathcal{B}G$. This has a ...
3
votes
1answer
104 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
3
votes
0answers
31 views

Total space of a geometric vector bundle

Suppose that $X$ is a scheme with a $B$-action, and that $\lambda: B \to \mathbb{C}^*$ is a character of $B$. We then have a geometric vector bundle $\pi: X \times^B k \to X/B$, where $B$ acts on $X ...
5
votes
1answer
93 views

What Information/Advantage do we Gain by Substituting a Continuous Map by a Fibration?

I'm trying to understand the usefulness of "substituting" a continuous map f , by a fibration F. By substituting, I mean there is the result that given a continuous map $f:X \rightarrow Y $ , for ...
3
votes
0answers
30 views

Steenrod Algebra as automorphisms of additive group

Is there a direct way to see that the subalgebra of the mod-$p$ Steenrod algebra ${\mathcal A}_p$ generated by the reduced powers is isomorphic to the dual of the Hopf algebra ${\mathcal ...
4
votes
0answers
116 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
4
votes
0answers
95 views

is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
4
votes
1answer
77 views

Inverse question: Recognizing when a phenomenon behaves like an algebra

I might be phrasing this question incorrectly, but my students asked me about it and I did not have a good answer. I am in statistics and I look at all sorts of social data, network data, climate ...
13
votes
5answers
382 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
1
vote
0answers
58 views

Fundamental group of a complete intersection in real projective spaces

I'm trying to understand the fundamental group of the following complete intersection in $RP^2 \times RP^2 \times RP^1$: \begin{eqnarray} &&t_1 \left( x_1^3 + x_2^3 + x_3^3 + a x_1 x_2 x_3 ...
6
votes
0answers
284 views

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$-manifold $X$ is a ...
1
vote
0answers
24 views

Simple question on splitting of cohomology groups.

From the exponential exact sequence, I have $$ 0 \rightarrow H^2(X,\mathbb{C})/H^2(X,\mathbb{Z})\rightarrow H^2(X,\mathbb{C}^\times) \rightarrow Tor(H^3(X,\mathbb{Z})) \rightarrow 0. $$ for some ...
2
votes
0answers
78 views

Short exact sequence of group of Hodge classes

I'm from a foreign country, I don't speak well English. Sorry. My question is : $X$ and $Y$ are subvarieties of a smooth projective variety $M$ such that $M=X \bigcup Y$. I would like to know if we ...
1
vote
1answer
66 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...
8
votes
0answers
82 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
2
votes
0answers
61 views

Max Noether's theorem application

I'm trying to solve this problem that I've found on the Internet related to Max Noether's theorem [AF+BG theorem (also known as Max Noether's fundamental theorem)] . It uses the notation of Fulton's ...
3
votes
0answers
53 views

Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i ...
2
votes
2answers
110 views

Genus of a curve: topology vs algebraic geometry

In topology one defines the genus $g$ of a connected orientable topological manifold $X$ as: The maximum number $g$ of cuttings along non-intersecting closed simple curves without rendering the ...
1
vote
1answer
67 views

relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
2
votes
1answer
100 views

Rick Miranda exercise complete intersection curve. Prove it and find genus.

The book by Rick Miranda asks to prove that the curve in $\mathbb{P}^3$ defined by the two equations $x_0x_3=2x_1x_2$ and $x_0^2+x_1^2+x_2^2+x_3^2=0$ is a smooth complete intersection curve. Also asks ...
1
vote
0answers
63 views

group law of complex torus is divisible?

I need help with this exercise: Show that the group law of a complex torus (the definition I have is that of Rick Miranda's book Algebraic curves and Riemann surfaces, the one that he constructs from ...
0
votes
0answers
209 views

Homeomorphism between complex torus and S1 x S1

We have the lattice L={$m_1w_1 + m_2w_2 | m1,m2 ∈ \mathbb Z, w1,w2 ∈ \mathbb C $}. We want to construct an homeomorphism between $\mathbb C/L$ and $\mathbb S^1 \times \mathbb S^1$. I've read that the ...
13
votes
1answer
238 views

What is the homotopy type of the affine space in the Zariski topology..?

I'm asking this question out of curiosity, as I was unable to come to a conclusion. Consider the affine space over an algebraically closed field, for concreteness we can work with $\mathbb{C}^{n}$. ...
3
votes
0answers
73 views

Euler characteristic of affine space

Sorry for the trivial question.. but what is the (topological) Euler characteristic of $\mathbb{A}^n$? Also, is there a genus-degree formula for affine curves similar to $g={d-1\choose 2}$ for smooth ...
1
vote
0answers
98 views

Genus of Fermat Curve

The genus of a projective Fermat curve $x^d+y^d=z^d$ in $\mathbb{P}^2$ can be computed using the formula $g={d-1\choose 2}$, where $d$ is the degree. Is the genus of the affine curve $x^d+y^d=1$ the ...
2
votes
1answer
89 views

Lift of a diffeomorphism of the Torus

I'm trying to prove the following formula. Suppose to have $p:\mathbb{R}^{d}\rightarrow\mathbb{T}^{d}$ the canonical projection of the real d- dimensional space in to the d-dimensional torus, and ...
3
votes
0answers
55 views

What is a $c_1$-map for Riemann-Roch theorems?

Atiyah and Hirzebruch define a $c_1$-map to spell out Riemann-Roch theorem for (compact and connected) smooth manifolds. The definition is following: a map $f:Y \to X$ is called a $c_1$-map if we are ...