Questions tagged [algebraic-topology]

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

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minimal genus of a $2$-homology class. Is it “natural” with respect to maps in some sense?

I was wondering if we have some results on "naturality" of the genus of an homology class. More specifically: Let $[C] \in H_2(X;\Bbb Z)$ be an homology class represented by an orientable submanfold $...
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Singular homology but only considering linear simplices $[v_0,…,v_n]$ is isomorphic to singular homology?

There is a homology theory of open sets of $\mathbb R^n$ defined as singular homology but instead of considering all maps $\sigma:\Delta^n \rightarrow U$ as simplices, we only consider the linear ones ...
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1answer
28 views

Homology of spheres: how to show that $\tilde{H}_{n-1}(S^{n-1})\cong\tilde{H}_{n}(S^{n})$, without Mayer Vietoris

I am searching for a proof of this fact, which is used for instance, in showing that the reflection of $S^n$ has deegre $-1$. Rotman proves it through Mayer-Vietoris, but in my professors' notes this ...
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33 views

Tammo tom Dieck's Algebraic Topology - van Kampen theorem for groupoids

For a topological space $X$, the fundmental groupoid $\Pi(X)$ has points of $X$ as objects and homotopy classes of paths from $x$ to $y$ as morphisms $x\to y$. A continuous map $f\colon X\to Y$ ...
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How to construct additional open sets in topology

Suppose X is a locally compact, connected space (in R, N, or Q). As your know, the topology usually studies the relationships between two or more spaces, but I have been curious about examining just ...
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25 views

Homology class of limit of surfaces

Let $(M,g)$ be a compact, connected and oriented Riemannian $3$-manifold with nonempty boundary. Suppose a sequence $\{S_n\}_{n \geq 1}$ of compact, connected, oriented and properly embedded surfaces (...
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2answers
47 views

Inclusion is a group isomorphism

I'm pretty new to topology and I'm having trouble solving some exercises since I'm not familiar with the notation used. First of all I'm not entirely sure what $\pi_1(X, p)$ means. My guess would be ...
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When is the inclusion of the constant paths $X\hookrightarrow X^I$ a cofibration?

Let $X$ be a space and $X^I$ the space of continuous maps $I=[0,1]\rightarrow X$ in the compact-open topology. There is a map $c:X\rightarrow X^I$, $x\mapsto c_x$, which sends a point $x\in X$ to the ...
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26 views

Normalization of curve and integral cohomology

Let $X$ be a 2-dimensional complex manifold, $C\subset X$ be a compact, reduced irreducible divisor. let $\nu: \tilde{C}\to C$ be a normalization(In this situation,$\tilde{C}$ is a riemann surface.)....
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46 views

Non-Naturality of the Splitting in the Universal Coefficient Formula

I want to show the non-naturality of the splitting in the universal coefficient formula for homology. The s.e.s. is $$0\to H_q(X,X';R)\otimes_R N\to H_q(X,X';N)\to Tor^R_1(H_{q-1}(X,X';R),N)\to 0$$ ...
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The $(i,j)$-entry of the matrix for $g_∗$ will be nonzero iff $g$ takes simplex $i$ to simplex $j$

Regarding the answer in the following link: Why is the trace of such a simplicial map is zero? (The answer gives great background for the question). I asked @John Palmieri a follow up question and ...
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What is the surface obtained by identifying antipodal points of $\mathbb{S}^1 \times \mathbb{S}^1$?

This is perhaps a soft question. Let $X=\mathbb{S}^1 \times \mathbb{S}^1$. Let $\mathbb{Z}_2$ act on $X$ by setting $(-1) \cdot (\theta,\psi)=(\theta+\pi,\psi+\pi)$. Consider the quotient space $X/ \...
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1answer
27 views

Lift of a continuous map $f:Y\to X$ to a covering space $Z$ given that loops in $Y$ lift to loops in $Z$

Suppose $X, Y$, and $Z$ are topological spaces. If necessary, you may assume that they are nice (manifolds). I am looking for the following result to be true even without this assumption however, so ...
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1answer
21 views

Polyhedral complex without ambient space

I would like to know the most commonly used terminology for the following simple object in combinatorics/topology. I restrict to the two dimensional case since this is what I am mainly interested in: ...
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1answer
32 views

Intersection Number of Stable and Unstable Manifold for Hyperbolic Fixed Point of ODE?

Let $Q^n$ be a closed, Riemannian manifold, $TQ$ its tangent bundle with the canonical lift of the Riemannian metric (as outlined in do Carmo) and the resulting compatible triple, $\xi$ a complete ...
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What is $H^i(\mathbb{RP}^n,\mathbb{RP}^{i-1};\mathbb{Z}_{2k})$?

I computed $H^i(\mathbb{RP}^n,\mathbb{RP}^{i-1};\mathbb{Z}_{2k})$ when $1\leq i$ in the two following ways: I consider $H^i(\mathbb{RP}^n,\mathbb{RP}^{i-1};\mathbb{Z}_{2k})\cong H^i(\mathbb{RP}^n/\...
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Why we are not allowed to turn rubber band inside out in obvious way?

Why we are not allowed to turn rubber band (easy visualization of inside and outside of circle) inside out in easy and obvious way (turning rubber band to flat annulus then turning to the other side) ...
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37 views

Problem in Proof of Theorem $5.3$ Chapter $2$ of “Cohomology of Groups by Kenneth S.Brown”

I am reading proof of Theorem $5.3$ in Chapter $2$ of "Cohomology of groups by Brown" on page number $42$. I have a problem with understanding the following consequence given in the proof: Let $G= F/...
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58 views

Homology is homotopy invariant, fake proof with simplicial methods?

Here is a maybe false proof that I came up with that homology of topological spaces is homotopy invariant. I'm thinking that it is indeed fake because why hasn't anyone else come up with this much ...
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1answer
74 views

Beginner Questions on CW-Complexes

As a beginner, I am struggling a bit with CW-complexes. I'm reading Hatcher, chapter 0. So I want to pose a few questions that are almost embarrassing to me but I believe it is important to ask such ...
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1answer
21 views

If a linear transformation maps each basis vector $e$ into the complement of the span of $e$, then its matrix has diagonal entries all zero

I'm currently reading the proof for Lefschetz fixed point theorem in Hatcher's Algebraic topology page 179. The conclusion in the last lines of the proof relies on the fact that If a linear ...
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Applications of higher topos in geometry and topology

Higher topos and derived algebraic geometry are relatively new areas and probably there are fewer people working on them compared to the majority of topologists or geometers. I haven't found geometers ...
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1answer
50 views

2 dimensional torus bundles over $\mathbb{S}^2$

With my limited knowledge of bundles, it seems that the isomorphism classes of principal torus bundles are in one-to-one correspondence with the homotopy classes of maps $[\mathbb{S}^2,\mathbb{C}P^\...
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1answer
54 views

Confusion about homology with coefficients in a ring $R$ and with coefficients in a $R$-module $M$

I used Universal coefficients theorem a lot, but now it seems to me that I have never understood it. My problem is the probably misunderstanding of homology with arbitrary coefficients. For what I ...
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1answer
39 views

Pullback bundle of bundle from real projective space to complex projective space

Let $V$ be complex vector space. As we know there is canonical map $\mathbb P(V_\mathbb{R})\rightarrow\mathbb P(V)$. Furthermore, let $H(V)=\{(\ell,v) : v \in\ell\}$ and we have a bundle $H(V)\...
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1answer
22 views

Spectral sequence with field coefficients

In the situation of the Serre spectral sequence for a fibration $F \rightarrow E \rightarrow B$, when can I say that the cohomology of $E$ with coefficients in a field is the direct sum of the ...
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1answer
45 views

Why is the group of units in a valued ring a topological group?

Let $A$ be a ring and $v$ be a valuation on $A$ with value group $\Gamma$. The open sets are: $U=\{a \in A:v(a)< \gamma\}$ for $\gamma \in \Gamma$. How should I show that the group of units ...
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1answer
28 views

Bounding $2$-simplexes in punctured plane in $\Bbb R^2 - \{0\}$ intuition

From Rotman's Algebraic Topology concerning developing the intuition on homology functors: The question we ask is whether a union of $n$-smplexes in $X$ actually is such a boundary. Consider the ...
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24 views

Genus and arithmetic genus?

In the Wikipedia of Arithmetic genus, for a complex projective manifold of complex dimension 1, the arithmetic genus satisfies $\chi=1-g$. I also see the relation $\chi=2-2g$ for a connected, ...
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26 views

Formula for juxtaposition of paths defined by equivalence classes?

I am interested in the equations for the juxtaposition of two paths $\alpha$ and $\beta$. I have the following equivalence relation; \begin{equation} (x,y) \sim (x',y') \iff x-x', q-q' \in \{0,1\} \...
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1answer
32 views

How do I calculate the cellular boundary map for $S^2$ with standard CW complex structure?

I'm using Hatcher's Algebraic Topology, and he gives: Let's give $S^2$ the standard CW complex structure consisting of two $0$-cells: $\{e^0_1, e^0_2\}$, two $1$-cells: $\{e^1_E, e^1_W\}$, two $...
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1answer
116 views

Is Swiss cheese homeomorphic to a ball?

In topology, I understand how most solids are homeomorphic to $n$-holed donuts, but I've never seen anyone mention internal holes (or should I say "bubbles", because holes are something else). If you ...
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45 views

Reference needed for Novikov paper

I'm having trouble finding this construction. Novikov (1964) constructed homotopy equivalences $f:N\rightarrow S^p\times S^q$ for $p,q > 1$ which are not homotopic to homeomorphisms. References? -...
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1answer
45 views

p-adic Pontryagin class invariance

It is a well-known result due to Novikov that the universal Pontryagin class of a tangent bundle over $\mathbb{Q}$ is a topological invariant. Also, it is known that over $\mathbb{Z}$ Pontryagin ...
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Does there exist a self-diffeomorphism of the disk with no conformal points?

This question is related to this one, though is supposed to be easier. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Does there exist a smooth volume-preserving diffeomorphism $f:D \to ...
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72 views

Is a solid sphere with a “bubble” in the middle topologically the same as a torus?

Suppose there was a solid sphere or ball with radius 2 such that a sphere with radius 1 was removed from the center making a hollow cavity. A sphere with a "bubble in it" if that's easier to visualize....
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1answer
24 views

Proving continuity for composed homotopy

Suppose $F$ and $G$ are homotopies between $f:X\rightarrow Y$ and $g:Y\rightarrow Z$, respectively. How to conveniently show that $$H(x,t)=G(F(x,t),t)$$ is continuous? I know how to do this recursing ...
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1answer
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If $A/B$ is homeomorphic to $C/D$, where $B\subset A$ and $D\subset C$, then is $H_i(A,B)=H_i(C,D)$?

On pg 125, in his book "Algebraic Topology" Hatcher makes a claim that is analogous to the following: If $A/B$ is homeomorphic to $C/D$, where $B\subset A$ and $D\subset C$, then $H_i(A,B)=H_i(C,D)$. ...
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1answer
44 views

Embedding into Adjunction space.

Let $X\cup_f Y$ be an adjunction space and $q:X\coprod Y\rightarrow X\cup_f Y$ be the associated quotient map. Let $A\subseteq Y$ be closed and $f:A\rightarrow X$ be a continuous map (this is called ...
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1answer
14 views

Disjoint union open and closed sets.

Suppose $(X_i)_i$ is an indexed family of non-empty topological spaces. Recall: $\coprod_{i\in I}X_i = $ $\{$ $(x,i)$ $:$ $x\in X_i$ and $i\in I$ $\}$ . There is a canonical injection $\sigma_i: X_i \...
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1answer
39 views

Is the chain map linear?

Simple question, but if $f_n$ is a chain map between $A_n$ and $B_n$ is $f_n$ linear as well? Since $f_n$ induces a homomorphism between homology groups, the induced map must be linear.
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1answer
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Coefficient dependences of cohomology group of spaces

In Hatcher's textbook, the cohomology groups of spaces are defined by the homology of the cochain complex Hom$_R(C_i(X;R),G)$, dual of the original chain complex $C_i(X;R)$ where $R$ is a principal ...
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1answer
40 views

Cup products in CW complexes (Hatcher, Example 3E.6)

Let $X$ be obtained from $S^2 \times S^2$ by attaching a $3$-cell to the second $S^2$ factor by a map $S^2 \to S^2$ of degree $2$. Then from cellular cohomology it follows that $H^*(X;\Bbb Z)$ ...
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2answers
48 views

Adjunction space quotient properties.

Let $X\cup_f Y$ be an adjunction space. Let $q:X\coprod Y \rightarrow X\cup_f Y$ be the associated quotient map, where $\sim$ is generated by $a\sim f(a)$ for all $a$. Show that $q$ is injective. My ...
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1answer
26 views

Bijection between basepoint-preserving homotopy classes and homomorphism between homological groups

I have a question on Exercise 3.1.13 on Hatcher's textbook: Let $\langle X,Y\rangle$ denote the set of basepoint-preserving homotopy classes of basepoint preserving maps $X\rightarrow Y$. Show that ...
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What does the notation $\pi_n(X) = 0$ and $\pi_n(X) = 1$ mean for a homotopy group?

For any path-connected space $X$, the set $\pi_n(X)$ is the set of homotopy equivalence classes of continuous functions $f:S^n \to X$ (some authors use loops instead). $\pi_n(X)$ is also a group under ...
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1answer
40 views

Cell structure of sphere with 3 points identified

I am trying to find the fundamental group of a sphere with 3 points identified. It is homotopy equivalent to a sphere wedged sum with two circles, so its group is the free group of two generators. ...
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57 views

Showing $H_3\left (\mathbb{RP}^4\right ) = \mathbb{Z}_2$

I previously took time to compute the homology classes for $\mathbb{RP}^2$ and $\mathbb{RP}^3$ with $$ H_k\left (\mathbb{RP}^2\right ) \;\; =\;\; \begin{cases} \mathbb{Z}, & \text{if} \; k =0 \\ \...
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1answer
52 views

Different complex structures on the real plane

I’m interested in studying how many possible different complex structures there are on the real plane. With complex structure, I mean any element of the quotient of the set of Riemann surfaces with ...
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1answer
51 views

Is this map from $(S^1)^n$ (n-copies of $S^1$) into $S^1$ continuous?

Let $S^1$ be the $1$-dimensional sphere given by $S^1= \{e^{i\theta} \ | \ 0 \leq \theta < 2 \pi \}$. Define a map $f:(S^1)^n \longrightarrow S^1$ given by $$f(e^{i \theta_1}, \ldots ,e^{i \...

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