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

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Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
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93 views

how to prove Euler Characteristic of cw complex $\chi(X)=\chi (A)+\chi (B)-\chi (A \cap B)$.

If a finite CW complex $X$ is the union of sub complexes $A$ and $B$, show that $\chi(X)=\chi (A)+\chi (B)-\chi (A \cap B)$. some how I can imagine what is happening,it is counting numbers of all ...
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24 views

The product of $E_2$-degenerate spectral sequences also $E_2$-degenerates?

Assume the Leray spectral sequence of a map $f_i:X_i\rightarrow B_i$ $E_2$-degenerates for $i=1,2$. Is it true that the Leray spectral sequence of the map $f_1\times f_2:X_1 \times X_2 \rightarrow B_1 ...
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1answer
52 views

Compatible notion of degree for $z^k : S^1 \to S^1$, when $k$ is not an integer?

I'm thinking about degree as the induced map on the first homology groups - the degree of $z^k$ is $k$, when $k$ is an integer. What happens when $k$ is not an integer? Is there a compatible notion of ...
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1answer
101 views

Topological degree of a complex valued map defined over a circle

Given a continuous map $f \colon S^n \to S^n$, it induces a map $f_{*} \colon \tilde{H}_n(S^n) \to \tilde{H}_n(S^n)$ of the form $f_{*}(z)=k*z$, where $k$ is an integer. Define the degree of $f$ as ...
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1answer
160 views

Why do these geometric assumptions imply these statements about relative homology?

I'm reading the paper Coverage in sensor networks via persistent homology. As in the paper, let $\mathcal{D}$ be a bounded domain in $\mathbf{R}^d$. We make the following assumptions: A5 The ...
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1answer
70 views

Soft Question: Geometry of Rings

So.. I was thinking about something last night and literally have no idea where to start. Can the ring axioms be connected to topology? If so, how? Basically, the motivation for my question has to ...
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1answer
111 views

Open covers by simply connected sets and fundamental group

I have a set $X$ which is path connected and it have an open cover by sets $U$ and $V$ which are simply connected, I am looking for a reference that shows that $\pi_1(X)$ is the free group with number ...
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0answers
84 views

In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators?

Consider homology and cohomology of some space $X$ where the homology groups are finitely generated. Consider $tor(H^i(X))$, the torsion part of $H^i(X)$. How do the generators of $tor(H^i(X))$ ...
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1answer
134 views

Homology subgroups generated by non-intersecting cycles

Suppose I have a closed genus $g$ surface. I can pick a canonical homology basis for the surface by picking $g$ "A-cycles" $a_1,\ldots,a_g$, and then $g$ "B-cycles" $b_1,\ldots,b_g$, represented by ...
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144 views

Homology of real projective space… I'm not satisfied with the argument in hatcher.

In example 2.42 Hatcher computes the homology of real projective space. I follow his argument, but I would be uncomfortable believing the details of the degree computation if I didn't see it in his ...
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1answer
38 views

Why is $S_{\ast}\left(X,A\right)$ free? [duplicate]

Why is $S_{\ast}\left(X,A\right)$ free? it is the quotient of two free groups $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$
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2answers
425 views

Homology of knot complement

I was told in a topology class that if $Y$ is a closed $3$-manifold and $K$ is a null-homologous knot in $Y$, then $H_1(Y- \nu(K)) \cong H_1(Y) \oplus \mathbb{Z}$. I'm trying to prove this ...
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0answers
53 views

Why the Objects of Homotopy Category not Homotopy Classes of Spaces?

A homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. I wonder why the objects are spaces, instead of homotopy ...
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1answer
45 views

Help proving a space is closed in order to show a space is properly discontinuous

This stems from exercise 6, section 81 in Munkres. Let $X$ be a locally compact Hausdorff space; let $G$ be a group of homeomorphisms of $X$ such that the action of $G$ is fixed-point free. Suppose ...
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94 views

extending maps from spaces to their whitehead towers

Let $f \,: X \to Y$ be a map between connected spaces. Let: $$ X^{(k)} \to \ldots \to X^{(0)} \approx X $$ and $$ Y^{(k)} \to \ldots \to Y^{(0)} \approx Y $$ be whitehead towers for $X$ and $Y$. What ...
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1answer
108 views

notation used in algebraic topology [closed]

i have some confusion in notations used in my algebraic topology class. $\approx$ homeomorphic $\simeq$ homotopy $\cong$ isomorphic Please correct me for the above.
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2answers
526 views

Does the Euler characteristic of a manifold depend upon the field of coefficients?

Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold ...
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2answers
46 views

various definition of connected

I am a beginner of AT and I cannot distinguish what we mean by 'connected', that is connected locally connected path connected could anyone help?
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25 views

what is the natural map from BP to $HF_p$?

what is the natural map from BP to $HF_p$? BP is the Brown-Peterson spectrum and $HF_p$ is Eilenberg Maclane spectrum. I am trying to learn the connections between ASS and ANSS. This map should ...
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0answers
79 views

Explicit expression for 1-forms which produce, by exterior derivative, a given 2-form in DR-cohomology for twice punctured plane

I want to check a possible solution to a problem I cannot solve: We know that closed 2-forms for $R^2-\{p,q\}$ are exact. Given a closed 2-form, what is a 1-form that gives us that 2-form under ...
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2answers
119 views

Calculate the Wu class from the Stiefel-Whitney class

The total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the total Wu class $u=1+u_1+u_2+\cdots$: The total Stiefel-Whitney class $w$ is the Steenrod square of the Wu class $u$: ...
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2answers
91 views

Why doesn't $S^n$ embed into $R^n$?

It seems obviously true, but how does one actually show this? Or what tools does one use? I only know the basics of homotopy theory and homology. Can I use invariance of domain somehow? If $S^n$ ...
3
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1answer
72 views

What is the maximal torus in the Lorentz group $O(m,n)$?

I'm close to certain it's just the product of the maximal tori of $O(m)$ and $O(n)$, but I can't quite prove it. I've tried the following: ...
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1answer
49 views

invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
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117 views

Classification of closed surfaces

I am doing a course in topology and is currently working on the classification theorem for closed surfaces. After realizing that every closed surface is either homeomorphic to the sphere or the sphere ...
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218 views

Homology groups of 2-disc with two circles cut out and their clockwise oriented boundaries identified…

Edit: Nevermind, I believe I found all of my mistakes, so I just need to redo the computation. I am working on Hatcher 2.2.9 c) He asks us to find the homology groups of 2-disc with two circle cut ...
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45 views

differential in AHSS for spin cobordism

According to these solutions, the differential $d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})$ is the dual of $Sq^2$. Why? This MO post asks a similar question (but about $d_3$ in ...
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1answer
146 views

When does a homotopy in $\mathbb{C}$ extend to a homotopy in the Riemann sphere?

I have a polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \ldots + a_0$, and I extend it continuously from $p : \mathbb{C} \to \mathbb{C}$ to a map from $S^2 \to S^2$. $(n > 0)$. In the plane, I have ...
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46 views

third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$

It is known that the third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$. Each colouring $C$ of a knot diagram by a dihederal quandle $R_3$ induces a homomorphism $f:C \rightarrow ...
3
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1answer
116 views

Are two spaces obtained from homeomorphic spaces by removing a ball still homeomorphic?

I have a specific example in mind. Consider $S_1,S_2$ two surfaces. Remove two discs to obtain surfaces with boundary $S_1',S_2'.$ If $S_1 \cong S_2,$ does it necessarily follow that $S_1' \cong ...
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2answers
44 views

Prove sequence has limit in $\gamma (S^1)$

This is a seemingly interesting exercise from my topology notes, but I can't solve it for the life of me. It's like this: take a closed curve $\gamma : S^1\to \mathbb R^2$, and a sequence ...
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55 views

How to prove the zeroth homology in the long exact sequence of associated to (A,X,X/A) …

(I am working out of Hatcher. This is theorem 2.13.) I am brewing up some confusion about the long exact sequence of the homology groups of $A \subset X$ and $X / A$. (For (X,A) a good pair, which is ...
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1answer
125 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
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0answers
172 views

Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$. ...
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2answers
63 views

fibered knots in $ S^3$

Given a fibered knot $k$ in $S^3$, we have the decomposition of $S^3$ as union of $M$ and $S^1\times D^2 $, where M is a fiber bundle over $S^1$, with fiber $F$ such that its boundary is the knot $k$. ...
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1answer
79 views

How do topologists count infinite dimensional holes?

For example, it seems like there "should" be an infinite dimensional hole (or perhaps many) in $S^1 \times S^1 \times \ldots$. (Or perhaps none...) Is there an invariant that would count it? What ...
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2answers
124 views

Universal cover of a torus “pillow”

I was thinking today, what is the universal cover of a torus with the "donut hole" shrunk to a point? I am certain it must include a sphere, but that can't be enough because of the point at the ...
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1answer
74 views

Computing quotients of abelian groups

Suppose that $A \cong \oplus_{i = 1}^{n} Z_{p_{i}^{k_{i}}}$ is some finite abelian group, and $(a_1, a_2, \ldots a_n)$ generates a subgroup $N$. If $\langle (a_1, a_2, \ldots a_n) \rangle$ was a ...
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2answers
71 views

How to compute the fundamental group of $S^2 / A$, where $A$ is a finite set of points?

Along the way to a much simpler solution to a homology problem, I thought about computing the fundamental group of $S^2 / A$. I quickly ran into trouble, so I want to know if there is there a slick ...
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37 views

Can a factor map be a Serre fibration?

Let $D_n$ be an $n$-disc. Is the factor map $p: D_n\to D_n/S^{n-1}\simeq S^n$ a Serre fibration, in other words, can any homotopy $F: [0,1]\times X\to S^n$ be lifted to $\tilde{F}: [0,1]\times X\to ...
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54 views

How to show $\frac{X\sqcup_{S^1}D^2}{D^2}\simeq \frac{X}{S^1}$

Let $X$ be a $CW$-complex which contains $S^1$. How to show $X\sqcup_{S^1} D^2/D^2$ is homeomorphic to $X/S^1$? Here $D^2$ is two dimensional closed unit disc in $\mathbb R^2$. My Atempt: Let ...
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1answer
51 views

Does ''homology vanishes eventually'' imply ''homotopy vanishes eventually''?

Let $X$ be a connected CW complex. Assume there is an integer $N\geq 0$ such that the singular homology $H_n(X)=0$ vanishes for all $n\geq N$. Is there an integer $M\geq 0$ such that $\pi_m(X)=0$ ...
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1answer
85 views

Do pi_1-surjective maps of degree 0 exist?

A well-known theorem asserts that degree 1 maps induce surjections of the fundamental group. I am looking for a partial converse. Is it true (under suitable assumptions) that a map between compact, ...
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1answer
61 views

Unit or non-zero octonions form an $A_\infty$-space?

If I have a Moufang loop, can it have a classifying space? I'm thinking of the unit octonions, if that's too general, so perhaps a better question is: are the unit (or non-zero) octonions an ...
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2answers
78 views

Calculating fundamental group and homology group to prove not homeomorphic to each other

Prove that $S^1\times S^3,S^2\times S^2,S^4,S^1\times S^1\times S^1\times S^1$ are not homeomorphic to each other by using fundamental group or homology group. I have known that the fundamental group ...
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1answer
57 views

Proof that homotopy equivalent maps induce equivalent homomorphisms on homology groups…

I am reading Hatcher, theorem 2.10. ( http://www.math.cornell.edu/~hatcher/AT/ATpage.html page 112 ) I mostly understand the proof, but am having trouble verifying that (except for the two), the ...
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1answer
62 views

Show a particular $f:S^{2n}\to S^{2n}$ must be antipodal map.

The following is a question from a qualifying exam I am studying for: Let $G$ be a group of homeomorphisms acting freely on $S^{2n}$. Show if $G$ has order $2$, then the nontrivial element must be ...
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1answer
122 views

Commutative Diagrams and their relationship to induced homomorphisms

Throughout my topology class my professor has used commutative diagrams on various occasions to prove results such as 1) There exists no antipode preserving, continuous, onto map, $f: S^2 \to S^1$ ...
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1answer
89 views

The fundamental group of a scheme / variety

On Wikipedia (http://en.wikipedia.org/wiki/%C3%89tale_fundamental_group) it's been written In algebraic topology, the fundamental group π1(X,x) of a pointed topological space (X,x) is defined as ...