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

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Why is $\mathbb{R}^2/G$ homeomorphic to the Klein bottle?

Let $G$ be the group of transformation generated by $a,b:\mathbb{R}^2\to \mathbb{R}^2$ where $a(x,y)=(x+1,y-1)$ and $b(x,y)=(x,y+1)$. We note than $bab=a$ and that $G$ acts properly discontinuously ...
6
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2answers
92 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
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1answer
34 views

A simplicial complex with vanishing first homology but nonzero fundamental group

I'm interested in the simplicial complex and I do not know much of algebraic topology to use the answers that question: ...
4
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0answers
23 views

All tangential Stiefel-Whitney numbers of $\mathbb{R}P^q$ are zero iff $q$ is odd, reference?

Where can I find a reference to the proof of the following fact? All tangential Stiefel-Whitney numbers of $\mathbb{R}P^q$ are zero if and only if $q$ is odd. I made a quick search through ...
4
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0answers
20 views

Composition of fibrations, fiber homotopy equivalence?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. If $f$ is a homotopy equivalence, does it necessarily follow that $f$ is a fiber homotopy ...
4
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0answers
43 views
+50

Passage to fixed point spaces is object function of a contravariant functor?

Let $X$ be a $G$-space. What is the easiest way to see that that passage to fixed point spaces, $G/H \mapsto X^H$, is the object function of a contravaraint functor $X^{(-)}: \mathscr{O}(G) \to ...
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1answer
23 views

almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
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0answers
47 views

Prerequisistes for P. May's A Concise Course in Algebraic Topology

I wonder what are the prerequisites for studying P. May's A Concise Course in Algebraic Topology. I understand basic point set topology and category theory are required. How much algebra does one need ...
5
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0answers
26 views

Based space, commuting in diagram up to homotopy.

Theorem. For any based space $Z$, the induced sequence$$\dots \to [Z, \Omega F f] \to [Z,\Omega X] \to [Z, \Omega Y] \to [Z, Ff] \to [Z, X] \to [Z, Y]$$is an exact sequence of pointed sets, or of ...
6
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1answer
60 views

$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)$

For Abelian groups $\pi$ and $\rho$, what is the easiest way to see that $$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)?$$My idea is the use the natural isomorphism$$[X, K(\rho, n)] \cong ...
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1answer
37 views

The fundamental group of the connected sum of two copies of the real projective plane

How do I compute the fundamental group of the connected sum $X \mathop{\#} X$, where $X$ denotes the real projective plane? I'd like to use Van Kampen's theorem, but I have trouble visualizing what ...
2
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1answer
54 views

Why is the unit circle, $\mathbf{S}^1$, a deformation retract of $\mathbf{R}^2$ minus any point?

It is clear that $\mathbf{S}^1$ is a deformation retract of $\mathbf{R}^2\setminus\{0\}$ since we can consider the straight line deformation retract $H\colon (\mathbf{R}^2\setminus\{0\}) \times [0, 1] ...
4
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1answer
100 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
5
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1answer
61 views

$\pi_7(S^4)$ contains an element of infinite order.

Show that $\pi_7(S^4)$ contains an element of infinite order. Now, I know that I should probably use the Hopf bundle here somewhere. I also know that $\pi_3(S^7) = 0$. But I am stuck. Can anyone ...
9
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0answers
81 views

Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

There is a short argument using Zorn's lemma and the compactness of $[0,1]$, that shows every manifold must have maximal open simply connected subspaces. However, I am wondering if it is necessarily ...
2
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0answers
24 views

Fibration implies inclusion is based homotopy equivalence? [on hold]

If $p: E \to B$ is a fibration, does it follow that the inclusion$$\phi: p^{-1}(*) \to Fp$$specified by $\phi(e) = (e, c_*)$ is a based homotopy equivalence?
5
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1answer
50 views

Why is $H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)$?

As the question title suggests, what is the easiest way to see that there is an isomorphism$$H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)?$$
8
votes
1answer
297 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
4
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0answers
16 views

Why study the p-completions of a space?

Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or ...
3
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1answer
50 views

Geometric Interpretation of Chain Homotpy

Let $X$ and $Y$ be topological spaces. Two maps $f,g:X\to Y$ are said to be chain homotopic if for each $n$ we have a map $T_n:C_n(X)\to C_{n+1}(Y)$ such that ...
1
vote
1answer
36 views

What is the meaning of “Continuous Group ”?

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" I did not understand the meaning of this statement " The method is based on finding a nontrivial continuous group ...
2
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1answer
24 views

Coproduct of the homology coalgebra of the sphere

Let $S^m$ be the $m$-sphere and $H_*(S^m)$ be the homology coalgebra with field coefficient. Then what is the coproduct of $ H_*(S^m) $? For $x$ the generator of $H_*(S^m)$, does $$ \Delta_*x=0? $$ ...
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0answers
49 views

Covering Space Question

I recently encountered the following: Let $p:(E, e_0) \to (B, b_0)$ be a covering map. Assume that $p_∗(\pi_1(E, e_0)) \subseteq \pi_1(B, b_0)$ is a normal subgroup. If $e_1\in p^{−1}(\{b_0\})$, then ...
0
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0answers
29 views

Extensions and Pushouts using an exact sequence of sets

This might seem a strange way of doing things, that is, inventing a possible example (according to comments, there is no such thing as an exact sequence of sets), but let us try to make one for ...
0
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0answers
28 views

inclusion of homotopy fiber and induced map on homology group

Given a fibration $F \to E \to B$, under what circumstances does the inclusion of the homotopy fiber into $E$, $F \to E$, induce injections on homology? The specific case I'm dealing with involves the ...
15
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0answers
115 views

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has ...
0
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0answers
34 views

Can a linear projection of spheres be a torus?

Assume that we have two disjoint subsets $A_1, A_2 \in \mathbb{RP}^4$ that are both homeomorphic to the sphere $S^2$. Let $\pi$ be the linear projection with centre a point that does not lie on $A_1$ ...
44
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6answers
5k views

Intuition of the meaning of homology groups

I'm studying homology groups and I'm looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible ...
2
votes
3answers
128 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
0
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1answer
23 views

How many Vertices, Edges, Faces are there in these Diagrams?

Apologies for the really basic question, however, I don't really understand how to count edges/faces for these types of diagrams with vertices "identified". E.g. For Figure 3.11, Clearly $V=4$, but ...
2
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0answers
35 views

For any Subspace $A$ of a Path-Connected Space $X$, we have $H_0(X, A)=0$.

I recently learnt about relative homologies and am wondering if the following is true: Statement: Let $X$ be a path-connected topological space and $A$ be a non-empty subspace of $X$. Then $H_0(X, ...
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0answers
24 views

Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
6
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0answers
59 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
3
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0answers
28 views

How to calculate homotopy invariant winding number?

Consider a map $f:S^1\to U(1)=S^1$, since we know $\pi_1(S^1)=\mathbb{Z}$, which measures how many times the map "wind" around the circle. Given some explicit form of the function $f(\phi$), where ...
6
votes
2answers
336 views

About the Riemann surface associated to an analytic germ

I've taken a small course in Riemann surfaces, and there is one part that I still don't understand (and I've been unable to find a reference that explains this rigorously and in detail). It is about ...
0
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0answers
29 views

Classifying surfaces

Please help me with part (ii) (i). The polygonal symbol of a certain surface without boundary is $ xy^{-1}x^{-1}zwz^{-1}vyw^{-1}v^{-1}$. Identify the surface. What is the Euler characteristic? ...
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1answer
43 views

Trying to understand relative homology group

I'm reading about relative homology group but I'm having hard time in understanding this concept. So I was trying to find $H_1(D^n,S^{n-1})$, but I'm unable to solve this problem. Can someone give ...
1
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1answer
48 views

Integral homology of $S^{n-1}/\pi$, $H_*(S^{n-1}/\pi; \mathbb{Z}_p)$

Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\mathbb{C}^n$, $n \ge 2$. ...
2
votes
1answer
39 views

Why does there exist a deck transformation mapping here?

See Kevin Dong's answer here. Let $\widetilde{\gamma}$ be a path in $X_N$ based at $x_0^N \in X_N$ with $p(\widetilde{\gamma}) = \gamma$ a nontrivial path in $X$. Since the cover is normal, there ...
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0answers
132 views

How do I prove this surface is homeomorphic to the sphere

Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$. How can I prove that S is homeomorphic to the sphere? ...
2
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2answers
74 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
17
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1answer
289 views

Why would I define Alexander–Spanier cohomology?

I think I can motivate the definitions of simplicial, singular, de Rham, Čech, and sheaf (co)homology, more or less. I might want to understand bordism, and start by trying to understand ...
1
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1answer
60 views

Is every simply connected open subset of $\Bbb R^n$ contractible?

Question: Is every simply connected open subset of $\Bbb R^n$ contractible? I know the result is true for $\Bbb R^2$ because by the Riemann Mapping Theorem every simply-connected proper open ...
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0answers
16 views

Spaces of Labeled Complexes (Munkres)

The following is taken from Munkres' Algebraic Topology book. I tried to determine which spaces (e.g. Mobius Strip, Klein bottle, etc) these complexes are, but to no avail. I computed the Euler ...
3
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0answers
44 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h ...
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0answers
11 views

Given a triangulation (labeled complex), how do we determine the space?

Given a triangulation, how do we tell which space it is representing? My first idea would be to calculate the Euler Characteristic, but that would still leave some ambiguity, e.g. both the Projective ...
11
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1answer
137 views

Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?

Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h \in H\} \subset X;$$$X^H$ is the $H$-fixed point subspace of $X$. ...
1
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3answers
50 views

Requirement “closed under finite intersection” in Van-Kampen-Theorem

Given a topological space $X$, a point $x_0 \in X$ and an open cover $\mathcal{U}$ of $X$ of path-connected subsets containing $x_0$ which is closed under finite intersections, we have by Van-Kampen ...
18
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1answer
302 views
+200

Prove Euler characteristic is a homotopy invariant without using homology theory

I was flipping through May's Concise Course in Algebraic Topology and found the following question on page 82. Think about proving from what we have done so far that $\chi(X)$ depends only on ...
7
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1answer
165 views

Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold ...