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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4
votes
1answer
154 views

Fundamantal group of a regular covering space

Let $B$ be the space of figure $\infty$ (with $x$(the red circle) and $y$(the black one) as generators) and $E$ its covering space (in the picture below). let $P_{*}: \Pi(E,a) \to \Pi(B,b) $ be the ...
2
votes
0answers
376 views

Cup Product Structure on the n-Torus

We know that The $k^{th}$ homology of the n-torus $(S^1)^n$ is generated by $\bigotimes_{i\in I}\lambda_i$ where $\lambda_i$ generates the first homology of the $i^{th}$ copy of $S^1$ and $I\subset\{1,...
6
votes
2answers
438 views

Mayer-Vietoris Type Sequence For Pushouts

Pushouts in the category $\mathsf{Top}$ of topological spaces exist and under certain conditions are known as adjunction spaces. Rigorously, if is a diagram in $\...
4
votes
0answers
234 views

is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
2
votes
1answer
335 views

Two deformation retractions (onto $A$) are homotopic (rel $A$).

This is a question from Hatcher's Algebraic Topology (Chapter 0, Question 13): 13. Show that any two deformation retractions $r^0_t$ and $r^1_t$ of a space $X$ onto a subspace $A$ can be joined by ...
5
votes
1answer
215 views

Homotopy between two functions to a circle.

Suppose $f,g: X\to S^1$ are such that $f(x)\neq -g(x)$ for any $x\in X$. I need to construct a homotopy between these two functions. Now, the fact that $f(x)\neq -g(x)$ guarantees that there is always ...
1
vote
2answers
690 views

Non-homotopy equivalent spaces with isomorphic fundamental groups

I know that if two spaces $X,Y$ have the same fundamental group : $\pi_1(X,a) \cong \pi_1(Y,b) $ does not imply that they are homotopy equivalent. But I can't find an example. I was thinking about ...
3
votes
0answers
86 views

Local dimension of graph embedding

I am trying to find a way to characterize the dimension of the smallest space into which a (neighbourhood of) a graph $\Gamma = (V, E)$ may be embedded. Although in the end my goal is to identify what ...
0
votes
1answer
136 views

Intuition on Whitney–Graustein theorem

According to the Whitney–Graustein theorem, two regular curves are regularly homotopic if and only if their winding numbers are the same. Suppose I have a circular curve but with an extra loop so ...
7
votes
1answer
154 views

How do Chern classes behave under connected sums?

I often see people talking about Chern classes of manifolds like $\mathbb{CP}^2 \# \mathbb{CP}^2$, or $\mathbb{CP}^2\#\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$, where $\#$ denotes connected sum and the ...
6
votes
1answer
169 views

Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
1
vote
1answer
371 views

What is a 3D winding number?

This paper mentions the term "3D winding number". Its abstract says: "We develop a new formulation, mathematically elegant, to detect critical points of 3D scalar images. It is based on a topological ...
4
votes
1answer
83 views

Inverse question: Recognizing when a phenomenon behaves like an algebra

I might be phrasing this question incorrectly, but my students asked me about it and I did not have a good answer. I am in statistics and I look at all sorts of social data, network data, climate data,...
1
vote
0answers
74 views

Compactness and Poincare duality

I am reading Appendix B in Fulton's Young Tableaux about Borel-Moore homology. In particular, I'd like to understand why for compact manifolds the Borel-Moore homology groups are isomorphic to ...
1
vote
1answer
290 views

a Problem about Degree of Map between Spheres

When I read the book "Algebra Topology-A First Course", I find a problem. It is on the Page 97, Exercise (16.15). Problem We define $f$, $g\colon \mathbb{S}^{n}\to\mathbb{S}^{n}$ to be orthogonal ...
2
votes
1answer
270 views

Are there degree-1 maps from $S^2 \times S^3 \rightarrow S^5$ or from $S^5 \rightarrow S^2\times S^3$?

This is a question from a past qualifying exam I am stuck on: For a smooth map $f:M\rightarrow N$ between smooth, compact, oriented $n$-manifolds, the degree of $f$ is the unique integer $k$ such ...
0
votes
1answer
64 views

$GL(n,\mathbb{C})$ is semi-locally simply connected.

How can we show that $GL(n,\mathbb{C})$ is semi-locally simply connected.
0
votes
1answer
76 views

existence of double covering [duplicate]

Let $M$ be a manifold , and $\pi_1(M)=\mathbb{Z}$. then can we say, the double covering of $M$ exists and is unique?
2
votes
2answers
81 views

Loops in $RP^2$

We know that $\pi_1(RP^2)=Z_2.$ How do non-trivial loops in $RP^2$ look like? (If $RP^2$ is the upper hemisphere $D$ with antipodal points of $\partial D$ identified)
0
votes
1answer
87 views

Path or One-Dimensional Manifold

The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional ...
2
votes
2answers
138 views

Self-contained text on characteristic classes

I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and ...
4
votes
1answer
137 views

Orbit space of S3/S1 is S2

I'm having trouble finishing this homework assignment. I did the first part by showing that the orbits are invariant: every element from the same $(S^1(z_1, z_2) \in S^3/S^1)$ is mapped to the same ...
0
votes
1answer
167 views

Deleted join of simplicial complex

Give an example of simplicial complex $k$ such that the deleted join of $k$ is homeomorphic with $S^1\times [0,1]$, i.e: $$k^{*^2}_{\bigtriangleup}\cong S^1\times [0,1]$$
3
votes
2answers
130 views

Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
0
votes
1answer
26 views

Why $\operatorname{im} \delta_{-1}$ contains two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic function?

In the paper, on page 21, line 15-20. It is said that $B^0=\operatorname{im} \delta_{-1}$ is the one dimensional space containing two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic ...
6
votes
1answer
250 views

Fat geometric realization weakly equivalent to the usual one

Let $X$ be a simplicial set. Recall that we can associate to it a certain topological space, the geometric realization, given by $ | X | = \int ^{n \in \Delta} X_{n} \times \Delta _{n} \simeq (\...
3
votes
1answer
193 views

Verifying continuity of the deformation retraction of the mapping cylinder

Given $f : X \rightarrow Y$ I have the mapping cylinder $M_f = (X \times I) \sqcup Y / \sim$ where $I = [0,1]$ and $(x,0)_1$ is glued to $f(x)_2$ (I'd like to be pedantic for this verification so I'm ...
6
votes
2answers
341 views

Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
0
votes
1answer
103 views

Composites of homotopic maps are homotopic.

There is a statement, which says: "Composites of homotopic maps are homotopic." But there is something, what i don't understand: 1) if we got two homotopies: $F:X \times I \rightarrow Y $ and $...
2
votes
1answer
139 views

Are space of paths between two different points and space of pointed loops only homotopy equivalent? What about smooth case?

Let $X$ be a path-connected CW-complex and $x$, $y$ points in $X$. Any choice of a path between $x$ and $y$ provides maps (in both directions) between the space $L(x, y)$ of paths from $x$ to $y$ and ...
3
votes
2answers
129 views

Properties of $\pi_n$ from a category theoretical point of view

This will be a more open question. I would like to have a better understanding about how the homotopy group functors behave with different constructions such as limits or colimitis. Some examples: ...
2
votes
0answers
95 views

coproduct of base and fiber is weakly equivalent to total space of a fibration in stable model category

Let $C$ be a proper pointed model category such that for any $X, Y \in C$ the natural morphisms $$QX \coprod QY \to X \coprod Y \to RX \times RY$$ are all weak equivalences (here $Q $ and $R$ are ...
3
votes
1answer
134 views

mayer vietoris homework

Let $X=A\cup B\ \ A,B$ are open and $A\cap B$ is contractible. Prove that $H_i(A\cup B)\equiv H_i(A)\oplus H_i(B)$ for $i\geq 2$. I think about using Mayer Vietoris sequence but I don't know how to ...
4
votes
1answer
238 views

A (somewhat) conceptual proof that the boundary of a fundamental class of a manifold with boundary goes to a fundamental class?

In this set-up, let M be a compact n-dimensional manifold with boundary $\partial M \neq \emptyset$. Assume that M is orientable, and that $[M] \in H_n(M,\partial M;R)$ is the fundamental class of M....
1
vote
3answers
79 views

Differential closed form

Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form: $$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$ in some ...
4
votes
0answers
751 views

Fundamental group of CW-Complex only depends on 2-Skeleton

I was just about to write down my answer to an exercise in algebraic topology and I wanted to use the fact that $\pi_1 (X)$ only depends on the $2$-Skeleton of $X$ for any $CW$-Complex $X$. I am very ...
5
votes
1answer
318 views

What are simplicial topological spaces intuitively?

(NOTE: I reposted the question to MO. Please answer there.) I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial sets are ...
0
votes
2answers
107 views

The singular homology and cohomology of a topological space with coefficients in a zero characteristic field.

I have a field with zero characteristic, like $K=\mathbb{C},\mathbb{R}$ and I want to show that the homology groups and cohomology groups with coefficients in these fields satisfy: $$H_n(X,K) \approx ...
5
votes
2answers
490 views

Definition of star in a simplicial complex

Given a simplicial complex K and a collection of simplices S in K, the star of S is defined as the set of all simplices that have a face in S. Now consider the following picture (from wikipedia): ...
0
votes
1answer
209 views

Cohomology groups of $K(\mathbb{Z}_p \times \mathbb{Z}_p,1)$

I have a question regarding the cohomology groups of the Eilenberg-MacLane space $K(\mathbb{Z}_p \times \mathbb{Z}_p,1)$. For $n$ > $2$, is there a way to show that $H^n(K(\mathbb{Z}_p \times \mathbb{...
12
votes
2answers
553 views

Motivation for introducing algebraic topology?

What kind of topological questions does algebraic topology answer where point set topology is not enough? Phrased differently: Where is the line (or maybe intersection) between point set topology ...
5
votes
2answers
237 views

Good source for a point set topological introduction to CW complexes?

Most algebraic topology books I found don't dwell too much on point set topology of CW complexes. I'd like too become more familiar with them. Anyone knows a good source (with exercises) too learn ...
2
votes
1answer
454 views

Torus minus disk does not retract

I'd like to show the following (intuitively clear) fact: Given a torus $T^2$ and an embedded disk $D\subset T^2$ (put a disk in the middle of the square whose edges we identify to get the torus), ...
3
votes
0answers
187 views

Classification of circle bundles over a 2-manifold with boundary

I want to understand and try to give a proof of the following claim: Let $B$ be a compact, connected topological $2$-manifold (surface) with nonempty boundary, then the $S^1$-bundles over $B$ with ...
2
votes
1answer
151 views

Real projective plane: $f_*$ isomorphism $\implies f$ surjective

Suppose we have a continuous map $f: \mathbb{R}P^2\rightarrow \mathbb{R}P^2$ that induces an isomorphism in homology $f_*: H_p(\mathbb{R}P^2) \rightarrow H_p(\mathbb{R}P^2)\ \ \ \forall p$. How do I ...
1
vote
0answers
21 views

Looking for a basic reference on propagators (in Topology)

I am looking for a basic (preferably self-contained) reference where I can read about propagators (as they appear in Topology), and in particular Morse propagators. Thanks!
13
votes
1answer
179 views

Can torsion in the fundamental group happen in “the real world”

Suppose that $X$ is a CW-complex such that $\pi_1(X)$ has non-trivial torsion. Does this imply that $X$ cannot be embedded in $\mathbb{R}^3$? Intuitively, this seems like it should be clear, since (...
5
votes
2answers
2k views

Fundamental group of projective plane is $C_{2}$???

I just recently know that there are topology with finite nontrivial fundamental group (homotopy curve). I just can't wrap my mind around it at all. If you have a curve, and somehow cannot shrunk it ...
1
vote
0answers
62 views

Restriction of a line bundle to a a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold $C$ can be characterized completely by its ...
2
votes
1answer
359 views

Taking off the vest while the jacket is still on

My topology professor said that: One can take off his vest while the jacket is still on. Is there a topological answer to this question? Of course there are some ground assumptions like : 1- ...