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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1answer
35 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 ...
0
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0answers
10 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
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1answer
41 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
52 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
109 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" ( ...
2
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0answers
41 views
+50

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
115 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
28 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?
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0answers
30 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)\}? ...
-4
votes
0answers
218 views

polynomial cohomology

Hope this finds you all well. I want to make sure of one thing : Do we usually have polynomial cohomology only in case the cohomology modules are free of rank 1 at most in each degree? PS:I don't ...
2
votes
1answer
33 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
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0answers
46 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
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1answer
21 views

Is collapsibility a homotopy invariant?

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

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

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
votes
1answer
291 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
43 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 ...
4
votes
0answers
57 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
59 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
53 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
112 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
57 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
votes
1answer
65 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
votes
2answers
84 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 ...
11
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2answers
157 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
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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
164 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
357 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
21 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
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
votes
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 ...
0
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0answers
22 views

Prove winding number is the same as index of a vector field. [on hold]

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
votes
1answer
78 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
115 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
887 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
votes
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 ...
0
votes
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 = ...
1
vote
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
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 ...
0
votes
0answers
44 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 ...
3
votes
1answer
32 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
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
votes
1answer
79 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
votes
1answer
388 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 $$ ...
0
votes
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!
0
votes
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
53 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 ...
1
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0answers
21 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 ...