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

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3
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1answer
50 views

cohomology ring of $S^2$ $\times$ $S^4$ and $CP^3$.

I was studying Hatcher's algebraic topology book.In page number 251,book says $S^2$$\times$ $S^4$ and $CP^3$ has same cohomology groups but they have different ring structure.I understand that they ...
-1
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0answers
44 views

Intuitive explanation of bordism invariant?

As the question suggests, what is an intuitive explanation of "bordism invariant"? I tried looking up some examples but they were drenched with jargon...
0
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2answers
66 views

Bisection Theorem

I don't know whether this is true or false But I did try to prove it as true using similar arguments as in the Bisection theorem Statement: Given a simple bounded region in $\mathbb{R}^2$ there exist ...
1
vote
2answers
64 views

Show that every simplicial complex is a CW-complex.

Klaus Janich's Topology, Page 97: The decomposition of a simplicial complex into its open simplices is a CW-decomposition. No further explanation of this proposition is provided after that,so ...
3
votes
1answer
65 views

Degrees of maps $RP^{2n+1}\rightarrow RP^{2n+1}$

What degrees are possible for maps $\mathbb R \mathbb P^{2n+1} \rightarrow \mathbb R \mathbb P^{2n+1}$? I'm asking about the odd dimensions because we cannot define degree (in a way that would make ...
3
votes
2answers
34 views

Map induced in mod $2$ cohomology of a projection $S^n \to S^n/\mathbb{Z}_2$

Consider the involution $\varphi_i \colon S^n \to S^n$ given by $(x_0, \ldots, x_n) \mapsto (x_0, \ldots, x_{i-1}, -x_i, \ldots, -x_n)$, where $0\leq i\leq n$. Let $f_i \colon S^n \to ...
1
vote
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 ...
3
votes
1answer
58 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
votes
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 ...
0
votes
2answers
57 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
votes
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
votes
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 ...
2
votes
1answer
69 views

Associativity of the smash product on compactly generated spaces

Given pointed topological spaces $X$ and $Y$, their smash product is the space $$ X \land Y = \frac{X \times Y}{X \times \{ y_0\} \cup \{x_0\} \times Y}, $$ where $x_0$ and $y_0$ are the distinguished ...
3
votes
3answers
211 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 ...
2
votes
1answer
104 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
votes
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 ...
2
votes
1answer
41 views

Does taking mapping spaces with a connected space preserve disjoint unions?

Let $X$ be a connected topological space and $\{Y_i\}_{i\in I}$ a family of spaces. Since the image of a connected space is connected, we obtain a natural bijective map ...
1
vote
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 ...
2
votes
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
votes
0answers
25 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 ...
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 ...
0
votes
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 ...
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 ...
5
votes
1answer
130 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" ( ...
0
votes
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)\}? ...
1
vote
1answer
33 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?
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: ...
1
vote
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
vote
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
votes
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 ...
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 ...
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?
1
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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), ...
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 ...
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
3answers
107 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 ...
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 ...
11
votes
2answers
165 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?
6
votes
1answer
129 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 ...
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 ...
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 = ...
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 ...
3
votes
1answer
33 views

localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$

Let $X$ be an $H$-space. Let $F$ be a field. Then $H_*(X;F)$ is a Hopf algebra over $F$. According to group-like elements in the Hopf algebra of the homology of H-spaces, $$ \pi_0(X)=\{g\in ...
2
votes
0answers
30 views

Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
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 ...