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

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0
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2answers
63 views

homology group of adjunction space

I start to study homology theory and i want to understand homology groups of adjunction space In this picture i can't see $V$ deformation retracts to $X$ neither intuitively nor explicitly help ...
3
votes
1answer
59 views

Is there a generalization of Jordan curve theorem? Not in higher dimension, but in the plane?

Jordan curve theorem (bit generalized one) Let $C_1$ and $C_2$ be closed connected subsets of $S^2$ whose intersection consists of two points. If neither $C_1$ nor $C_2$ separates $S^2$, then ...
2
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2answers
54 views

Another clarification about Thom-Pontrjagin construction

This is the second part of the following solved question. [I'm following Bredon's Book]. After explaining the idea behind the "desired" bijection we want to build, Bredon start dealing with the ...
1
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0answers
32 views

Is every boundary of a connected subset of $S^2$ whose closure does not separate $S^2$ a Jordan curve?

Let $C$ be a nonempty connected subset of $S^2$ and $\partial C$ be the boundary of $C$. If $\bar C$ (closure) does not separate $S^2$, is $\partial C\cong S^1$? This seems very true, but I ...
5
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3answers
108 views

What Topics of Mathematics to study to go into Big Data [closed]

I am interested in Big Data and related jobs after graduation from Math PhD/Masters, what topics and fields of Mathematics should a student learn that are most relevant to Big Data? Currently, I know ...
2
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1answer
74 views

If the product of two homotopy equivalences is a homotopy equivalence are the factors homotopy equivalences?

The question says it all: Given two maps $f\colon A\rightarrow B$ and $f'\colon A'\rightarrow B'$, such that their product $$f\times f'\colon A\times A'\rightarrow B\times B'$$ is a homotopy ...
0
votes
1answer
69 views

How can I write Klein bottle as an adjunction space?

I want to find the homology groups of the Klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout $S^1\cup_f D^2$ but I ...
2
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1answer
44 views

Signs in definition of $\Delta$-complexes

In Hatcher's book nth chain group defined as free module with basis consist of n-symplexes in $\Delta$-structure. Consider some basis element in $\Delta_1(X)$: $$\sigma:[v_0,v_1]\to X$$ Our simplex ...
0
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2answers
59 views

Closed sets in Spec(k[X,Y])

On page 74 of Mumford's red book (attached) it is stated that a proper closed set in Spec(k[X,Y]) is composed of finitely many irreducible curves and finitely many closed points. Why does such a union ...
4
votes
0answers
47 views

Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$

I would like to compute the first Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$ in terms of the generators of $\mathbb C\mathbb P^2$ and $\overline{\mathbb C\mathbb P^2}$. ...
0
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0answers
28 views

Relationship between simplicial complex and abstract simplicial complex

We have the following definitions: A simplicial complex is a set of simplices so that any face of a simplex is a simplex and the intersection of two simplices is a face of both. An abstract ...
2
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1answer
105 views

Text book for sheaf theory

Is there any nice text book for sheaf theory for an under gradute student? Tennison's sheaf thory was too hard for me, Please help me, Thanke you very much.
2
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0answers
45 views

Chern-Gauss-Bonnet theorem for even-dimensional manifolds with boundary

On the wikipedia page for the Chern-Gauss-Bonnet theorem it states that there is a generalization of the theorem for even-dimensional manifolds with boundary, but does not provide the relevant theorem ...
3
votes
3answers
213 views

Examples of manifolds that are not boundaries

What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds? Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional ...
1
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0answers
24 views

Leray-Hirsch theorem for cohomology modulo torsion

Suppose $X,Y$ are smooth manifolds, $H^*(X,\mathbb{Z})$ is finitely generated. (*)Why do we have isomorphism modulo torsion: $H^n(X\times Y,\mathbb{Z})=\oplus_{p+q=n}H^p(X,\mathbb{Z})\otimes ...
0
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0answers
26 views

To use topology and Riemann surface in number theory.

I would like to learn some rudiment topology and Riemann surface in order to apply in number theory. I already know some algebraic topology, like covering space and fundamental group, singular ...
1
vote
1answer
32 views

Property of contractible spaces

Suppose $X$ and $Y$ are two complex tori. Let $V$ and $V'$ be the tangent spaces at identity. Then the exponential map, which we denote by $\pi:V\longrightarrow X$ and $\pi':V'\longrightarrow Y$ are ...
1
vote
1answer
56 views

Induced map on the homology

Although there are good articles about this theme like induced map homology example, I would like to get a more explicit answer. I know that one way to find such a map is the following: $ f:X\to Y ...
2
votes
2answers
53 views

Stiefel-Whitney classes with Z-coefficients

Stiefel-Whitney classes are defined (for example, in Milnor's Characteristic classes) as elements of the cohomology groups with $\mathbb{Z}_2$-coefficients as follows. $$w_i(E)=\phi^{-1} Sq^i(u)$$ ...
5
votes
1answer
132 views

How to obtain Grothendieck’s “Long March Through Galois Theory”

Several works cite "La longue marche a travers la theorie de galois". The work by Leila Schneps "Grothendieck’s "Long March through Galois theory" ( ...
5
votes
0answers
77 views

How do the direct and inverse image sheaf functors interact with homotopy?

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for ...
2
votes
1answer
118 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
1
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1answer
34 views

Equivalence of branched covers of the Riemann sphere

Consider the functions $f(z)=z^4$ and $g(z)=z^4+1$, branched covers of $S^2$. These functions have the same branch data, so they should be equivalent in some way. In what way are they equivalent?
0
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0answers
32 views

isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? ...
2
votes
1answer
36 views

Homotopy of certain maps induced homotopies

Let $\psi$ be a homeomorphism and $\gamma :[0,1] \rightarrow \mathbb{R}^2$ a path. Now assumme additionally that $(\psi \circ \gamma)(t) \neq \gamma(t)$ everywhere. Then we can look at the map ...
2
votes
0answers
49 views

General topology exercise (equivalent condition for simple connection)

Let $X$ be a pathwise-connected topological space. Prove that $X$ is simply connected iff every continuous $f:S^1\to X$ can be extended to a continuous function $g:D^2\to X$. How can I use the fact ...
1
vote
1answer
24 views

Is collapsibility a homotopy invariant?

There are some similar characterizations of a simplicial complex, with the implication relations: ...
8
votes
1answer
296 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
1
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0answers
62 views

Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb ...
5
votes
0answers
61 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
1
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1answer
66 views

Trying to make sense of this proof in Hatcher

So I'm trying to understand this proof in Hatcher's Algebraic Topology. Lemma: The composition $\Delta_n(X)\xrightarrow{\partial_n} \Delta_{n-1}(X)\xrightarrow{\partial_{n-1}}\Delta_{n-2}(X)$ is zero ...
2
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0answers
61 views

Literature concearning characteristic classes?

It sounds the literature about characteristic classes is not very abundant (am I wrong?). Whenever I look for books dealing with this matter I'm always lead to the same material like the classical ...
4
votes
1answer
114 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
3
votes
1answer
67 views

Clarification about the Thom-Pontrjagin construction as explained in Bredon's book

In Bredon's book, at page 118-119, there is a little chapter about the Thom-Pontrjagin construction, and I'm trying to follow the reason depicted there. He starts with a map $f \colon R^{n+k}\to ...
5
votes
1answer
69 views

Functorial properties of the compact open topology.

Let $X,Y,Y'$ be topological spaces and $A\subseteq Y$ a subspace. Every set of continuous maps is equipped with the compact-open topology. Is the canonical map ...
2
votes
1answer
37 views

Computation with Mayer-Vietoris and fundamental classes

Let $M$ be an $n$-dimensional closed oriented connected manifold and suppose that $\bar{U},\bar{V}\subset M$ are $n$-dimensional manifolds with boundary so that $M=\bar{U}\cup \bar{V}$ and ...
11
votes
2answers
166 views

Hodge Theory, intuition?

We have the following theorem of Hodge, as follows: $$\dim \ker \Delta^p = \dim H^p(M) = b_p(M).$$My question is, what is the intuition behind this statement?
0
votes
0answers
30 views

Can a fiber bundle with fiber R and structure group Z_2 be considered a vector bundle?

I have a principal $\mathbb{Z}_2$-bundle, then by Borel construction I get a fiber bundle with fiber $\mathbb{R}$ and structure group $\mathbb{Z}_2$. Can I say that this is a vector bundle?
3
votes
0answers
175 views

On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
2
votes
3answers
362 views

Showing higher homotopy groups of $S^1$ are trivial

I'm trying to prove $\pi_{i} (S^1) \cong 0$ if $i>1$. Is this correct. You have a short exact sequence, $\mathbb{Z} \rightarrow \mathbb{R} \rightarrow S^1$ (from the fiber bundle of the covering ...
1
vote
0answers
22 views

Calculation of Fixed point set

Let $p,q$ be two distinct primes and let $G_p$ and $G_q$ be nontrivial $p$-group and $q$-group respectively. Let $G = G_p \rtimes G_q$. Embed $G_q \subset U(n_1), G \subset U(n) $ where $U(n_1), ...
2
votes
1answer
74 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 ...
4
votes
1answer
99 views

Map to $RP^2 \vee S^1$ nullhomotopic

Let $R$ be $S^{1}\vee S^{1}$. Call the first circle by $a$ and the second one by $b$. Let $X$ be space by attaching two $2$-cells to $R$ one via the boundary map $a^{3}$ and the other via the boundary ...
6
votes
1answer
130 views

Is “Categories and Sheaves” a good followup to Aluffi's “Algebra: Chapter 0”?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it ...
11
votes
1answer
906 views

Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?

May's A Concise Course in Algebraic Topology gives the following statement of the Seifert-van Kampen theorem for fundamental groupoids $\Pi(X)$ (section 2.7): Theorem (van Kampen). Let ...
0
votes
2answers
68 views

Extending a homotopy equivalence

I have a basic question regarding homotopy equivalence. Let $X$, $Y$, and $Z$ be three subsets of $\mathbb{R}^2$ such that $(X\cap Z)\subset (Y\cap Z)$ are homotopy equivalent, and $X\setminus Z = ...
1
vote
1answer
64 views

What is a homology of chain complex?

First of all I don't know much about homological algebra, and algebraic topology I just took a class using Kinsey book (Topology of Surfaces) which is undergraduate math class and I learned what is ...
5
votes
1answer
58 views

What is the relation between cohomology of $K(\mathbf{Z}_2, n)$ and $K(\mathbf{Z}_2, n+1)$?

I'm reading a proof to a theorem, "Suppose $a, b: H^*(-, \mathbf{Z}_2)\to H^{*+k}(-, \mathbf{Z}_2)$ are two stable (commuting with suspension isomorphism) cohomology operation of degree k. If ...
3
votes
0answers
65 views

Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy ...
1
vote
1answer
44 views

Show $X$ is simply-connected given properties of two subsets

I'm given: $X$ is a manifold, $X = U\cup V$ for $U,V \subset X$ open, connected and simply-connected, with $U \cap V$ connected. And given this, I want to show $X$ is simply connected. Attempt I ...