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

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5
votes
1answer
138 views

Computing Homology using Mayer-Vietoris

(This is exercise 2.2.28 from Hatcher) Consider the space obtained from a torus $T^2$, by attaching a Mobius band $M$ via a homeomorphism from the boundary circle of the Mobius band to the circle ...
0
votes
1answer
55 views

How does one give topological structure to an abstract simplicial complex?

Given an abstract simplicial complex $K,$ I'd like to know how I can endow it with a topology.
6
votes
1answer
79 views

Intersection of two quadrics

How to understand (maybe, informally) why the intersection of two quadrics in general position in $\mathbb{CP}^3$ is an elliptic curve? It is obvious that it is a compact 2-manifold, i.e. a sphere ...
9
votes
2answers
195 views

What is the way to see $(S^1\times S^1)/(S^1\vee S^1)\simeq S^2$?

What is the way to see $S^1\times S^1/(S^1\vee S^1)\simeq S^2$? Even just an intuitive walkthrough. I can't visualize this quotient in my head.
2
votes
1answer
62 views

Gluing $2$-cells when viewing $S^2$ with antipodal points on the equator $S^1$ identified.

Consider the space $X$ which is $S^2$ with $x\sim -x$ for $x$ on the equator $S^1$. I was reading this answer here. When putting a cell structure on $X$, it says the two hemispheres are wound twice ...
2
votes
1answer
59 views

Total space of vector bundle deformation retracts onto 0-section of base space

I'm trying to prove the following: Total space of vector bundle deformation retracts onto 0-section of base space. Books seem to take this fact for granted. I checked Bott Tu and Hatcher. Online ...
2
votes
1answer
59 views

Showing $S^1$ is not a retract of $D^2$ using homotopy

I'd like to know if my argument below, in which I try to show that the 1-sphere is not a retract of the 2-disk using homotopy, is valid. Suppose there is a retract $r:D^2 \to S^1$. Then we can ...
0
votes
0answers
58 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
3
votes
0answers
30 views

Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
2
votes
1answer
81 views

De Rham cohomology for $\mathbb{R^2}$

De Rham cohomology groups for $\mathbb{R^2}$. $H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R}$ since $Z^{0}(\mathbb{R}^{2})$ is the one dimensional space of locally constant functions on $\mathbb{R}^{2}$ and ...
4
votes
2answers
76 views

Degrees of maps in algebraic topology

Please can I have some tips on how to construct maps between topological spaces of a given degree? For example, how would you go about building a map of degree 3 from CP1 cross CP2 to CP3? Or a map ...
5
votes
1answer
256 views

Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
5
votes
2answers
527 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
0
votes
1answer
31 views

Euler class is odd under orientation, thus its integral over a manifold will be even.

I learned a statement from others: "Euler class is odd under orientation, thus its integral over a manifold $M$ is even." I cannot fully appreciate it, can someone show this explicitly? The ...
2
votes
0answers
40 views

Notation and shorthand of cobordism G=O, SO, U

This is really a basic question: From Wikipedia: "The basic examples are G = O for unoriented cobordism, G = SO for oriented cobordism, and G = U for complex cobordism using stably complex ...
6
votes
3answers
137 views

Minimal Degree of map $S^2\times S^2\mapsto \mathbb{CP}^2$

I am having troubles finding the minimal d such there is a map of such degree from $S^2\times S^2\mapsto \mathbb P^2$. I know that the cohomology ring of $\mathbb P^2$ is thus I know that the degree ...
1
vote
1answer
86 views

Is unit circle a simplicial complex?

If so , please help me to find the simplices. I think it is homotopic to a point .so there is only one vertex.
1
vote
1answer
29 views

Avoiding self-intersections of immersed manifolds

Let $i: N\to M$ be an immersion of manifolds. If $\dim M\geq 2\dim N+1$ (or something like that?), does there exist arbitrary small perturbations of $i$ (wrt. some reasonable norm) that are already ...
1
vote
1answer
45 views

Prove that $H^{2}(S^{2})\neq 0$

Prove that $H^{2}(S^{2})\neq 0$ Suppose $\omega$ is an area form of $S^{2}$. An arbitrary two form on $S^{2}$ is closed as if $f(x,y)dx\wedge dy\in\Omega^{2}(S^{2})$ then $d(f(x,y)dx\wedge dy)=0$. I ...
4
votes
1answer
80 views

Degree of map using Poincare Duality

I have a very basic question. I want to compute the topological degree of the map $\phi: \mathbb{C}P^n \rightarrow \mathbb{C}P^n$ mapping $(z_0:\dots:z_n)$ to $(z_0^d:\dots:z_n^d)$, which, if life is ...
2
votes
0answers
45 views

lifting injective maps to injective maps in principal bundles

Let $i \, :Y \hookrightarrow X$ be an inclusion of (nice) topological spaces, and suppose that the induced map $\pi_1(Y) \to \pi_1(X)$ is injective. Then every lifting of $i$: $$ \tilde Y \to \tilde ...
1
vote
1answer
33 views

Show that a set defines a simplicial complex

Let $A_n := \left\{-n,\cdots,-1,1,\cdots,n\right\}$ $\Delta_n := \left\{ B \subseteq A \; \big\vert \; \#(\{-i,i\}\cap B)\leq 1 \; \forall 1 \leq i \leq n \right\}$ Show that ...
0
votes
0answers
54 views

Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
0
votes
0answers
37 views

what is $H_i(S^n-X)$ when $X$ is a subspace of $S^n$ homeomorphic to $S^k \vee S^l$ or to $S^k \sqcup S^l$.

Compute $H_i(S^n-X)$ when $X$ is a subspace of $S^n$ homeomorphic to $S^k \vee S^l$ or to $S^k \sqcup S^l$. it seems that I should use this:when $h:S^k \rightarrow S^n$ is an embedding then we have ...
2
votes
1answer
114 views

Determining the induced map on homology $\tilde{H}_n(\mathbb{R}^n-\{0\})$ of $f\colon \mathbb{R}^n\to\mathbb{R}^n$ based on sign of $\det(f)$.

I'm having difficulty understanding the following. It appears as Exercise 7, p. 155 in Hatcher's Algebraic Topology: (this is not homework, by the way) For an invertible linear transformation ...
2
votes
0answers
33 views

Degree One Map induces Surjections on Homology

Is the following statement true: If $f:M\to N$ is a degree one map of compact closed manifolds, then $f$ induces surjections $f^*:H_q(M)\to H_q(N)$. I found this claimed on ...
2
votes
1answer
53 views

Use Hurewicz Theorem to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$

Want to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$, using Hurewicz theorem. This is one of the questions on the previous topology qualifying exams. Any help will be appreciated! I am thinking in stead ...
2
votes
0answers
64 views

Degree of a map from $ T^2 \rightarrow T^2$

Given a map $f:T^2 \rightarrow T^2$, the degree of $f$ is given by the induced homomorphism $f^*:H_2(T^2) \rightarrow H_2(T^2)$. If I know the induced homomorphism $f^{**}:H_1(T^2) \rightarrow ...
2
votes
1answer
63 views

extension problem of a spectral sequence

From Hatcher's SSAT, If the coefficient group $G$ is a field, then $H_n(X;G)$ is the direct sum $\oplus_p E^\infty_{p,n-p}$ of the terms along the $n^\text{th}$ diagonal of the $E^\infty$ page. ...
1
vote
1answer
41 views

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 ...
1
vote
0answers
48 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 ...
1
vote
0answers
18 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 ...
0
votes
1answer
48 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 ...
4
votes
1answer
85 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 ...
2
votes
1answer
148 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 ...
0
votes
1answer
63 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 ...
3
votes
1answer
93 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 ...
0
votes
0answers
34 views

Fundamental group of a 2-sphere with a line segment from north pole to south pole. [duplicate]

Let X be the topological space consisting of the standard 2-sphere together with a line segment from the north pole to the south pole. Compute $\pi_1(X)$ and construct the universal covering ...
1
vote
0answers
66 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 $tor(H^i(X))$ ...
0
votes
0answers
19 views

How does one make cochain complex of sphere into an associative DGA?

Given the singular chain complex of the sphere $S^n$, $S^*(S^n)$, a reference says that one can use the Alexander Whitney product to make $S^*(S^n)$ into an associative differential graded algebra. ...
3
votes
1answer
103 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 ...
7
votes
1answer
76 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 ...
1
vote
1answer
33 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 $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$
4
votes
2answers
152 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 ...
1
vote
0answers
50 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 ...
3
votes
1answer
32 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 ...
1
vote
0answers
70 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 ...
0
votes
1answer
62 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.
7
votes
2answers
285 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 ...
0
votes
2answers
39 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?