Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

learn more… | top users | synonyms (1)

7
votes
0answers
73 views

How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
1
vote
1answer
30 views

Book recommendation: Homology and Cohomology

I need to learn some concepts about Homology and Cohomology theory to apply in riemannian geometry basically, but really I have not time to read about that. I know just two books of W. S. Massey, ...
0
votes
0answers
26 views

Can we recover homology from cohomology

The universal coefficient theorem allows one to calculate cohomology by homology. Can we recover singular homology by cohomology for a complex manifold? Can a complex manifold (algebraic manifold) ...
5
votes
3answers
663 views

Local homeomorphisms which are not covering map?

I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local ...
0
votes
0answers
20 views

Does the restriction of the Thom class of a submanifold to the cohomology of the submanifold give the Chern class of the normal bundle?

Let $X$ be a compact complex manifold of (complex) dimension $d$, let $i\colon Y\hookrightarrow X$ be a regular complex submanifold of (complex) codimension $k$, let $\mathcal{N}_{Y/X}$ be the normal ...
6
votes
1answer
632 views

Braid groups and the fundamental group of the configuration space of $n$ points

I am giving a lecture on Braid Groups this month at a seminar and I am confused about how to understand the fundamental group of the configuration space of $n$ points, so I will define some ...
2
votes
1answer
42 views

Covering spaces of $S^1 \vee S^1$

The question is: Let $x_0$ be the common point of two circles in $X = S^1 \vee S^1$. Let $a$ and $b$ be the standard generators of $\pi_1(X, x_0) = \langle a, b\rangle$ corresponding to the two ...
3
votes
1answer
27 views

Homology and Neighborhood

Let $X$ a connected manifold, $x \in X$ and $V$ a neighborhood of $x$. Assume $i:V \to X$ induce isomorphism between all homology groups. Does $X-p$ and $V-p$ still have the same homology groups ? ...
-1
votes
0answers
35 views

Triangulation of triangle

Why it's not triangulation(named in Hatcher as $\Delta$-complex structure)?(edges are glueing)
0
votes
5answers
55 views

Finite set of points of $R^n$ is compact

In order to show that a finite set of points of $R^n$ is compact, I just need to show that the set is closed and bounded. First of all, since it's a finite set, I can Always pick the greatest ...
6
votes
1answer
47 views

A Ham Sandwich type problem

If $A_1,...,A_n$ are measurable subsets of $S^n$, then there is a great $S^{n-1}$ cutting each $A_i$ exactly in half. The tools I have at my disposal are the Borsuk Ulam theorem and the Ham ...
1
vote
0answers
54 views

Relative de Rahm cohomology computation for two disjoint circles embedded in $\mathbb{R}^2$

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
1
vote
1answer
21 views

van Kampen theorem for fundamental groupoid of $X$ relative to $A$

Let $X$ be a manifold with submanifold $A \subseteq X$. Let $\Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For ...
5
votes
2answers
174 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
2
votes
0answers
27 views

Analyzing the following space:

I recently encountered the following space: the underlying set is $C = C_1 \cup C_2$, where $C_i$ is the circle of radius i and centre 0 in the complex plane. Basic open sets are: • {z} for every z ...
2
votes
0answers
53 views

Compute the fundamental and homology groups of $S^3 \setminus K$, where $K$ is two linked copies of $S^1$ in $\mathbb R^3$

Compute the homology groups of $S^3 \setminus K$, where $K$ is two linked copies of circles in $\mathbb R^3$. How about the homology group of $S^3 \setminus K'$ where $K'$ is just one copies ...
5
votes
1answer
71 views

Is $f$ necessarily a homotopy equivalence? [closed]

Let $X = \{(p, q): p \neq -q \} \subset S^n \times S^n$. Define a map $f: S^n \to X$ by $f(p) = (p, p)$. My question is as follows: is $f$ necessarily a homotopy equivalence? EDIT: Here is my ...
0
votes
0answers
29 views

Derivative group action [duplicate]

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
2
votes
1answer
24 views

Find all surfaces that can be obtained from an octagon by identifying edges in pairs.

Find all surfaces that can be obtained from an octagon by identifying edges in pairs. I think there are many many surfaces. Can anyone give some hints for the question?Thanks.
1
vote
1answer
96 views

Some possible mistakes in Bott and Tu

I've been reading Bott and Tu's book "Differential Forms in Algebraic Topology" and it seems that this book has an enormous list of errors. Therefore I would like to confirm if the parts that I will ...
3
votes
1answer
254 views

Partition of Unity question

I am starting to read the book "Differential Forms in Algebraic Topology" by Bott and Tu. In the proof of the exactness of the Mayer - Vietoris sequence (Proposition 2.3, page 22 - 23) a partition ...
1
vote
0answers
34 views

Existence of Shafarevich maps(theorem 3.6) on Kollar 's book

I have some problem when reading Theorem 3.6 of Kollar's book Shafarevich Maps and Automorphic Forms, page 41 (Corollary 3.5 of this article ), which states that Let $X$ be a normal variety, ...
0
votes
0answers
20 views

recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
1
vote
0answers
30 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
5
votes
1answer
53 views

Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...
1
vote
0answers
28 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
2
votes
0answers
47 views

Fundamental group two torus minus a single point?

So, if I take one torus and take of one single point, what will be its fundamental group? I think that one single point will not change the topology in this sense. Or will? If yes, how?
0
votes
1answer
116 views

Suspension of a CW complex

I want to prove that the suspension $\Sigma X$ of a CW-complex $X$ is a CW-complex, buy I'm starting with CW-complexes and I don't have a clue of how start, so I'd appreciate any help. Thanks. ...
13
votes
2answers
1k views

What does the Hodge conjecture mean?

I read from the Internet that according to the Hodge conjecture, a certain harmonic differential form in a projective, non-singular algebraic variety is a rational linear combination of the cohomology ...
2
votes
0answers
51 views

when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
0
votes
0answers
24 views

Homotopy Equivalence and Local Coefficient Systems

Suppose I am computing the (co)homology of a nice space M using a local coefficient system G, i.e. $H_{*}(M, G)$. If M is homotopy equivalent to N, then M and N should have isomorphic (co)homology. ...
1
vote
0answers
50 views

Long exact homology sequence in singular homology

I am trying to understand/develop the proof of the following theorem: Let $R$ be a commutative ring with 1. Suppose $(C_*, c_*), (D_*, d_*), (E_*, e_*)$ are $R$-chain complexes and $i_*: C* ...
1
vote
1answer
57 views

Question about simply connected spaces.

I am reading Hatcher's Topology and in it, it is noted that a space is simply connected, by definition, if and only if it is path connected and has trivial fundamental group. Can someone provide some ...
0
votes
0answers
18 views

Definition of structure group associated with fiber bundle.

I was studying fiber bundle from Spanier book.In that book there is a definition of structure group associated with a fiber bundle.But I am not able to understand the definition properly.Could ...
0
votes
1answer
20 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
0
votes
0answers
54 views

Prerequisites “Homology Theory of Algebraic Varieties” by Wallace

I bought this book because the title was very interesting, the description as well and the price very cheap. You can read it here in PDF. Unfortunately, I realized after reading the first lines I was ...
2
votes
0answers
27 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
1
vote
1answer
29 views

The preimage of a curve in the projective plane by the quotient map.

Let $q:S^n \rightarrow \mathbb{R}P^n$ the quotient map between the $n$-sphere and the $n$-dimensional projective plane. Prove that if $\alpha$ is a curve in the projective plane then $p^{-1}\alpha$ ...
1
vote
1answer
46 views

Homology of $S^2/x\sim -x$ for $x$ on the equator

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim -x$ for $x$ in the equator $S^1$. Compute the homology groups $H_i(X)$. I wrote my solution/attempt below and I would like ...
2
votes
1answer
25 views

Why is the induced homomorphism an injection?

I am reading Hatcher's Algebraic Topology. One of the propositions says that if a space X retracts to a subspace A, the the homomorphism i# induced by the inclusion i: A --> X is injective. It is ...
2
votes
0answers
60 views

Singular homology: Change of coefficients

Let $f: X \to Y$ be a map of topological spaces which induces isomorphisms $H_*(f;\mathbb{Z})$ on singular homology with $\mathbb{Z}$-coefficients. Show that $f$ induces isomorphisms ...
1
vote
0answers
28 views

A function between covering spaces.

Given $p_1:\bar X_1\rightarrow X$ and $p_2:\bar X_2\rightarrow X$ covering maps. Proof that if exist $f:\bar X_1\rightarrow \bar X_2$ continuos and surjective then $f$ is a covering map. I don't know ...
2
votes
1answer
47 views

How can I get a cohomology of hypersurfaces by using their equation?

While studying about complex projective hypersurfaces, I attempts to find a cohomology of this hypersurface : $$X_n=\{(x_0:x_1:x_2:x_3) \in \mathbb{C}\mathbb{P}^3~|~x_0^n+x_1^n+x_2^n+x_3^n=0\}$$ I ...
1
vote
0answers
43 views

Function from $S^n$ on $S^1$.

Show that for $n\geq 2$ there is not any function $\phi: S^n \rightarrow S^1$ such that $\phi(-x)=-\phi(x) $ I have no idea about how to solve this problem. It is quite similar to Bursuk-Ulam ...
16
votes
1answer
252 views

When is there a submersion from a sphere into a sphere?

(Edit: Now posted to MO.) That is: For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$? The discussion at this math.SE question has narrowed it down to the ...
2
votes
0answers
29 views

Base of homology on a Riemann surface and holomorphic differentials

I have two questions: 1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of ...
2
votes
1answer
36 views

homomorphism of $H$-spaces between a monoid and loop space of its classifying space

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. The geometric realization ...
1
vote
0answers
30 views

spin structures definition

One way to define a spin structure on a $SO(n)$-bundle $p:E\to B$ is to require that there is an element $\sigma\in H^1(E;Z_2)$ such that restricted to each fiber gives a generator of ...
1
vote
0answers
21 views

Computing monodromy eigenvalues of a generic arrangement

First the set-up: Let $f = \Pi_{i=1}^{d} f_{i} \in \mathbb{C}[x,y]$ be a generic, homogoeneous hyperplane arrangement of degree d. Let M be the complement of $f^{-1}(0)$, $F = f^{-1}(0)$, and $p: ...
6
votes
1answer
122 views

Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$

Throughout, $(X, x_0)$ and $(Y, y_0)$ will be connected pointed topological spaces. If $f : (X, x_0) \to (Y, y_0)$ is a continuous map and $f_* : \pi_n(X, x_0) \to \pi_n(Y, y_0)$ is an isomorphism ...