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

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5 views

Is collapsibility a homotopy invariant?

There are some similar characterizations of a simplicial complex, with the implication relations: ...
0
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0answers
10 views

Cohomology group of pairs (X,A) where X is thickening figure 8 and A is its 3 boundary components

Calculate Cohomology group of pairs (X,A) where X is thickening figure 8 and A is its 3 boundary components using simplicial structure.(without using UCT for relative pairs) I have no idea how to ...
8
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1answer
290 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 ...
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0answers
16 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 ...
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0answers
35 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
48 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 ...
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0answers
48 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
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1answer
111 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
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1answer
52 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 ...
4
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1answer
62 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 ...
1
vote
1answer
31 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 ...
6
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2answers
81 views

What Topics of Mathematics to study to go into Big Data

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 ...
10
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2answers
146 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?
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0answers
28 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
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0answers
162 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
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3answers
356 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 ...
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0answers
20 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
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1answer
69 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 ...
12
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2answers
2k views
+100

Normal subgroups of free groups: finitely generated $\implies$ finite index.

I am looking at what should be a simple exercise in geometric group theory. I have reduced the problem to just completing an exercise from Hatcher, Section 1.B page 87: 7. If $F$ is a finitely ...
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0answers
20 views

Prove winding number is the same as index of a vector field.

I'm trying to prove that the winding number and the index around a point in a vector field are the same. I know that the index is sometimes defined as the winding number but I'm working with the ...
4
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1answer
65 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
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1answer
108 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
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1answer
879 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 ...
10
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2answers
390 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
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2answers
64 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 = ...
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1answer
62 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
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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 ...
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0answers
42 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
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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 ...
3
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1answer
30 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 ...
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0answers
27 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 $E^a$ ...
0
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1answer
78 views

What is a counterexample that a composition of covering maps not a covering map?

Let $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ be covering maps. What would be an example that $q\circ p:X\rightarrow Z$ is not a covering map? I saw a counterexample here, but it was too complex. Is ...
15
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1answer
387 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
4
votes
1answer
37 views

group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
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0answers
55 views

What does $D^n$ refer to?

I'm not sure what object $D^n$ is in the following exercise: "Write down an explicit homeomorphism between $D^n/S^{n-1}$ and $S^n$." Thanks!
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0answers
24 views

Difference between $\mathbb{Z}$-acyclic and $\mathbb{Q}$-acyclic

In page 1853 of Graham's Handbook of Combinatorics, Volume 1, it is stated that for simplicial homology, $$\mathbb{Z_p}\textrm{-acyclic}\Leftrightarrow \mathbb{Z}\textrm{-acyclic} \Rightarrow ...
2
votes
1answer
52 views

Computing Klein bottle's cohomology ring in $\mathbb{Z}$

Well I've been struggling with this one. This is the picture of the Klein Bottle. It has two triangles (U upper, V lower), three edges (the middle one is "c") and only one vertex repeated 4x. So my ...
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0answers
20 views

Do complete Hopf algebras have an antipode?

I am reading Quillen's paper on rational homotopy theory. In appendix A of this paper Quillen defines a notion he calls complete Hopf algebras. These are certain cocommutative bialgebra structures ...
3
votes
2answers
44 views

Example of induced homomorphism in algebraic topology

I would like to understand what induced homomorphism are, as they appear in the definition of the Mayer-Vietoris sequence. Since an homology group $\tilde{H}_n$ is a quotient group defined as ...
7
votes
1answer
93 views

different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
0
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1answer
28 views

With field coefficients homology and cohomology coincide

With field coefficients the universal coefficients theorem takes the form: $$H^n(X;F)=Hom_{F-modules}(H_n(X;F),F)$$. Now in all computations I have seen with field coefficients we have ...
3
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1answer
151 views

Cross product of cohomology classes: intuition

Let $X$ and $Y$ be topological spaces and consider cohomology over a ring $R.$ Hatcher (in his standard Algebraic Topology text) defines the cross product of cohomology classes $$H^k(X) \times ...
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1answer
23 views

Homology and Reduced homology coincide on non trivial pair.

In Hatcher page 118, he says that There is a completely analogous long exact sequence of reduced homology groups for a pair $(X;A)$ with $A\not = \emptyset$ ; This comes from applying ...
0
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1answer
379 views

Computing the homology of $\mathbb{S}^2$ via Mayer-Vietoris

I'm trying to compute the homology of the $2$-sphere. I start by decomposing the sphere into a northern hemisphere and southern hemisphere, denoted by $A$ and $B$, respectively, and allow these two to ...
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0answers
36 views

the functor Ext repairs the inexactitude of Hom on the right

For a short exact sequence of R -modules: $ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0$. Prove that: the following sequence is exact $0 \rightarrow \mathrm{Hom} ...
15
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1answer
208 views

Why would I define Alexander–Spanier cohomology?

I think I can motivate the definitions of simplicial, singular, de Rham, Čech, and sheaf (co)homology, more or less. I might want to understand bordism, and start by trying to understand ...
2
votes
1answer
66 views

Would this be a homology theory?

Consider a manifold $M$, and denote by $\Delta _p M$ the set of all submanifolds of dimension $p$ (with or without boundary) of $M$. Define $G_pM$ to be the free abelian group generated by $\Delta_p ...
4
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0answers
43 views

Minimum number of sets required for a good open cover

A good open cover of a topological space is an open cover such that all open sets in the cover, and all finite intersections of open sets in the cover are contractible. For example, $S^2$ has an ...
5
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3answers
115 views

On defining homology groups

I have been trying to understand what homology groups are "talking about," and now I am wondering if the following works as a definition of homology. But first, some illustration of what it is ...
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0answers
39 views

When is the Zariski closure of subset connected [closed]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is an indecomposable ring, how can we show that the ...