Tagged Questions

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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0answers
6 views

Augmentation Homorphism on an Algebra

I have asked the definition of augmentation ideal filtration and this question is also related. I am reading Combinatorial Group Theory in Homotopy Theory, I by Fred Cohen and here is a part of the ...
1
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1answer
16 views

Bijection of homotopy classes

I want to prove the following: given the (already proven) fact that the we have a bijection between (continuous maps) $f:X\rightarrow Y^K$ and $g:X\wedge K\rightarrow Y$ for pointed spaces $X$,$Y$ and ...
4
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1answer
26 views

Are $C^\infty$ exotic spheres $C^k$ exotic?

The only theory of exotic spheres that I know is of $C^\infty$ structures on them; that is, that there are plenty of spheres (in dimensions $n \geq 7$ that are homeomorphic but not diffeomorphic. To ...
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2answers
22 views

Possible degree of a cover $p \colon S^{2n} \to X$

I'm asked to compute all the possible degrees of a covering space $S^{2n} \to X$, where $X$ is a path connected space. My idea is to try to show that these degrees can only be $1$ (take the identity ...
4
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1answer
27 views

Integrality of the $L$-genus for a smooth manifold

For a smooth manifold $M^{4k}$, the Hirzebruch signature theorem gives the signature $\sigma(M)$ in terms of a polynomial $P_k$ in the Pontryagin numbers of $M$ whose coefficients are rational but ...
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1answer
12 views

Proving these spaces are homeomorphic

As part of a proof I am reading it states that the part of the sphere given by: $S$ = $\{(x,y,z) \in \mathbb{R} : x^{2} + y^{2} + z^{2}=1, x \geq 0, y \geq0, z\geq0\} $ is homeomorphic to the closed ...
0
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0answers
18 views

Homotopic attaching maps give Homotopy Equivalent spaces

I want to prove that if $f,g : S^{n-1} \to X$ are homotopic maps then the resulting spaces $X \cup_f D^n$ and $X \cup_g D^n$ are homotopy equivalent. I know this question has been asked before: ...
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1answer
23 views

Dependence of Euler characteristic on the coefficients

My question is similar to this one but I think it is different. Suppose we are given an infinitely generated free abelian group, which forms a $\mathbb{Z}_{2}$-graded chain complex, such that its ...
2
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2answers
33 views

Reference request for equality of torsion of H1 and H2

I have heard that for a surface $X$ (algebraic? smooth? compact?) the torsion part of $H_1(X,\mathbb{Z})$ is the same as that of $H_2(X,\mathbb{Z})$. Please could you give me a correct statement? I ...
1
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1answer
30 views

Illustration of homotopies

While there are many pictures path homotopies, I fail to find any that illustrate normal homotopies (in the event "a normal homotopy" is something else, I clarify that I mean given two continuous ...
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2answers
23 views

Fundamental group of surface of genus $g$

Suppose we have a compact surface of genus $g$. How to calculate its fundamental group ?
3
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0answers
16 views

examples of k-invariants of spectra

The homotopy groups of commonly used topological spectra (like KO, S, MO, MSO, etc) are easy to find in literature, even appearing on Wikipedia's List of Cohomology Theories; however, I have had some ...
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0answers
25 views

Show that both $*$ and $.$ operation are same.

Let $G$ be a topological group with identity $x_0$. Let $\pi_1(G,x_0)$ is a fundamental group with the usual $*$ operation. If we define $(f.g)(s)=f(s)g(s)$ $\forall s\in [0,1]$ $\forall f,g\in ...
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0answers
30 views

Question 13 in section 2.2 Hatcher Homology

Let $X$ be the 2 complex obtained from $S^1$ with its usual cell structure by attaching two 2 cells by maps of degrees 2 and 3 , respectively. (a) Compute the homology groups of all the subcomplexes ...
1
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1answer
11 views

Loop of a topological group acting on different points being homotopic to constant maps.

Let $G$ be a path-connected topological group that acts on the path connected space $X$. Let $\alpha$ be a loop in $G$ based at the identity, and let $x, y$ be points in $X$. I want to show that ...
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1answer
36 views

What is the definition of “Augmentation Ideal Filtration”?

Let $A$ be an algebra. What is the definition of the Augmentation Ideal Filtration of $A$? Any answer with reference will be greatly appreciated.
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0answers
44 views

Question about a solution to a problem in Hatcher

I'm reading through a solution to a problem in Hatcher and I'm not sure why this line is true. Could anyone explain?
2
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1answer
20 views

Abuse of notation in relative homology theory

I am refreshing my understanding of homology theory (well, recreating from scratch really!) after a thirty year break and there's something that bugs me in how the texts I've seen write about relative ...
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1answer
46 views

Prove that any two maps S^m → S^n, where m < n, are homotopic

Prove that any two maps S^m → S^n, where m < n, are homotopic. I've been fiddling around with trying to use the Simplicial Approximation theorem since that's the material we've recently covered in ...
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0answers
35 views

Prove that there are only countably many homotopy classes of maps |K| → |L|, where K and L are finite simplicial complexes [on hold]

Let K and L be finite simplicial complexes. Prove that there are only countably many homotopy classes of maps |K| → |L|.
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2answers
35 views

Fundamental group of countably many holes and one limit point

So I proposed this Fundamental group of a space of infinite genus and an accumulation point same question to my professor in Complex Analysis when we were going over the fundamental group and ...
0
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1answer
33 views

Fundamental group of $\mathbb{P}^n(\mathbb{C})$

We know that $$\Pi_1(\mathbb{P}^n(\mathbb{R})) \cong \mathbb{Z}_2 $$ for $n \geq 2 $. Is there a similar statement for $\Pi_1(\mathbb{P}^n(\mathbb{C}))$ ?
2
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1answer
29 views

A question about characteristic classes

I have a map $\phi:BO(1)^n\rightarrow BO(n)$ which is given by sending any $n$-tuple in $BO(1)^n$ to an $n$-plane through the origin. Thus, this induces a group action on the symmetric group $S_n$ ...
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0answers
20 views

What's the significance to the $m$ in the notation $L(n,m)$ for the Lens space?

I'm reading a quick example (Example 12.13 of Topological Manifolds by John Lee) of the construction of the lens space $L(n,m)$. Basically, let $$S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$$ ...
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1answer
69 views

isotopy of homeomorphisms of a torus [on hold]

Let's consider some homeomorphism of a torus which is isotopic to identity. Is it possible to construct an explicit isotopy? Edit: It's well-known statement that a homoemorphism of a torus is ...
1
vote
1answer
21 views

Homology groups of three faces with a point on the common edge removed

Consider this situation: There is an edge between two vertices, with three faces (maybe half-disks or half-squares, it doesn't really make a difference to topology as far as I know) going out from it, ...
1
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1answer
20 views

Path connected iff the action of $\pi_1(Y,y)$ on $p^{-1}(y)$ is transitive.

Let's say we are looking at a covering space $X$ of $Y$. Let $K \subset \pi_1(Y,y)$ be the subgroup of elements $$K:= \{ [\gamma] \in \pi_1(Y,y) : x * [ \gamma] = x\}$$ $K$ is the group of paths in ...
2
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1answer
43 views

Question regarding notation in algebraic topology

My class has not been following a book and my professor's last bit of notation is a bit confusing to me. This is the goal. We are given a path-connected space $Y$ and $H$ a subgroup of $\pi_1(Y,y)$. ...
1
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2answers
37 views

Why existence of universal covering implies that the base space be locally path connected?

I am reading Chapter 13, the chapter about classification of covering spaces, of J.Munkres' Topology. My confusion raised when I read Corollary 82.2. which says: the space $B$ has a universal ...
2
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1answer
35 views

Not all finitely-presented groups are fundamental groups of closed 3-manifolds

It is a well-known result that, for any finitely-presented group $G$ and any integer $n \geq 4$, there exists a closed $n$-manifold whose fundamental group is isomorphic to $G$ (a sketch of proof can ...
2
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0answers
27 views

The identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$

Show that the identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$. Here $\Delta^n$ is the n-simplex, and I know $\delta \Delta^n$ denotes its ...
2
votes
2answers
30 views

Is composition of covering maps covering map?

In Munkres book, composition of covering maps is covering map when $r^{-1}(z)$ is finite for each $z$ in $Z$ where $q : X\to Y$ , $r:Y\to Z$ are the covering maps. I tried hard to find an example that ...
0
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1answer
26 views

How to Prove that these Spaces are not Homotopically Equivalent

Let $X=\{0\}\cup\{1/n:n\in\mathbb{N}\}$ and $Y$ be any countable discrete space. Show that $X$ and $Y$ are not homotopically equivalent. The hint says: Every continuous map from $X$ to $Y$ takes all ...
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0answers
25 views

The generator of compact cohomology of punctured plane

Can anyone give detailed derivation of the generator of compact cohomology of $H^1_c (R^2-\{0\}$). (It is homotopic to a circle so it is isomorphic to $R$ but I want computation of its generator, ...
0
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1answer
30 views

What does “minus the zero section” mean?

I see a vector bundle obtained from another one by the means of "minus the zero section" in some literature. The concept zero section of a vector bundle is found in Section Sections and locally free ...
2
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1answer
43 views

the 0-th homology of a simplical complex

Let $K$ be an (abstract) simplicial complex. The claim is: $H_0(K;\mathbb{Z})$ is always nonzero. Is this possible to prove it without any "special techniques to computing homology-groups"? ...
2
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1answer
30 views

Help on formalisation proof of the triviality of a kernel in Mayer-Vietoris

Consider the Mayer-Vietoris sequence for $\mathbb{RP}^2$, where the two open sets are $U:= \{ [x;y;z] \in \mathbb{RP}^2 | z \neq 0 \}$ and $V = \mathbb{RP}^2 \setminus [0;0;1]$. I've proved that $U ...
3
votes
1answer
27 views

What is the $\pi_1$-action on the hom-sheaf between two finite etale covers?

Say you have two finite etale covers $X\rightarrow S$, $Y\rightarrow S$. The hom sheaf $\mathcal{H}om_S(X,Y)$ on the etale site $\text{Sch}/S$ is finite locally constant, hence representable by some ...
3
votes
2answers
31 views

How to compute the fundamental group of this space?

I know that without the closed disk, a sphere with the a diameter deformation retracts onto the wedge sum of a circle and a sphere. But I can't figure out how to deform the disk to a suitable ...
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1answer
35 views

Any invertible linear map is homotopic to a composition of reflections

I am trying to solve a problem in Hatcher. I reduced my problem to showing that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is an invertible linear map, then $f$ is homotopic to a composition of reflections, ...
0
votes
1answer
29 views

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$.

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$. $[X,Y]=\{f:X\to Y,f$ continuous $\}/\sim$ where $\sim$ is the homotopic equivalence. ...
0
votes
1answer
26 views

Prove that $C_1$ and $C_2$ are homotopic fixing endpoints.

Let $C_1$ and $C_2$ be two great circles in $S^2$, intersecting at the points $p,q$. If we consider $C_1$ and $C_2$ as curves starting and ending at $p$. Prove that $C_1$ and $C_2$ are homotopic ...
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1answer
43 views

Relation about Disk and Sphere

Definition of sphere and disk are following \begin{align} S^n =\{ (x_1 , \cdots x_{n+1}) \in \mathbb{R}^{n+1} | \sum x_i^2 =1 \} \end{align} \begin{align} D^n =\{ (x_1 , \cdots x_{n}) \in ...
4
votes
2answers
81 views

What categorical limits and colimits does $\pi_1$ preserve?

$\pi_1$ is a functor from the category of pointed topological spaces to the category of groups. It's clear that this functor preserves products, since $\pi_1(X\times Y)=\pi_1(X)\times\pi_1(Y)$, and a ...
1
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1answer
42 views

Dual notion of the subspace topology

Let $X$ and $Y$ be sets with $\iota:X\to Y$ an injection. If $Y$ is a topological space, we define the subspace topology on $X$ as the initial topology induced by this diagram. Analogously, if $X$ ...
2
votes
1answer
27 views

Homotopy groups of infinite Grassmannians

Let $G_k$ be the infinite rank $k$ Grassmannian. For $n>k$, is $\pi_n(G_k)$ trivial? Phrased differently, is every rank $k$ vector bundle on an $n$-sphere, for $n>k$, trivial? (This is motivated ...
1
vote
1answer
20 views

Proof explanation Mayer-Vietoris and the Punctured Euclidean Space.

I am only releasing part of the proof. Doesn't this prove that "otherwise" case is completely wrong? I am assuming "otherwise" refers to $n < 2$? Because he just showed that $H^0(\Bbb ...
3
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1answer
49 views
+50

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
1
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2answers
56 views

Why exactly is this injective? Algebraic Topology.

Let $(A^*, d^i)$ be a chain complex of finite dimensional vector spaces, i.e, $$0 \to A^0 \to A^1 \to \dots \to A^n \to 0.$$ Show the sequence $$0 \to H^i(A^*) \to A^i/Im(d^{i-1}) \to Im(d^i) ...
6
votes
1answer
52 views

Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?