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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2
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0answers
41 views

Closure relations of the cells in the Bruhat decomposition of the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
8
votes
3answers
444 views

Is there a non-simply-connected space with trivial first homology group?

Is there a path connected topological space such that its fundamental group is non-trivial, but its first homology group is trivial? Since the first homology group of a space is the abelianization of ...
2
votes
1answer
109 views

Simplicial Homology: The definition of cycles

I'm trying to convince myself beyond a doubt that $n$-cycles should be defined as elements of $\ker \partial _n$. My intuition is along the lines of "a cycle is a boundary of some chain (not ...
0
votes
0answers
57 views

Exactness of a certain sequence

Let $M$ be a connected manifold of dimension $m$ and let $\beta\in H^{1}(S^{1})$ such that $\int_{S^{1}}\beta = 1$. Let $\pi_{1}:M\times S^{1}\rightarrow M$ and $\pi_{2}: M\times S^{1}\rightarrow S^{1}...
0
votes
1answer
80 views

homology of suspension

Let $\Sigma$ be suspension. For any CW-complex, or topological space, does the reduced homology satisfy $$ \tilde H_*(\Sigma^k X)=s^k\tilde H_*(X)? $$ Here $s^k H$ is a copy of $H$ such that an ...
0
votes
1answer
30 views

Nullhomotopy special case.

If we have a $k$-dim proper smooth submanifold $N \varsubsetneq M$ and suppose we have the retraction map $r|_N:M→N$. Furthermore, we impose the condition that $N \cong S^n \times S^n$ (diffeomorphism)...
5
votes
5answers
172 views

Path connected but not metrizable

What are the examples of path connected spaces which are not metric spaces. The only examples I know are sets with indiscrete topology? Are there such spaces which are not simply connected (the ...
3
votes
2answers
88 views

Homology groups of $\mathbb{R}^3 - \{C_1,C_2\}$ where $C_i$ are disjoint circles

I am reviewing for my topology final and came up with this example. I want to compute the homology groups of $X = \mathbb{R}^3 - \{C_1,C_2\}$ where $C_1$ and $C_2$ are disjoint copies of $S^1$, so ...
1
vote
1answer
36 views

Nullhomotopy generalization.

If we have a $k$-dim proper smooth submanifold $N \varsubsetneq M$ and a continuous map $r|_{N } : M \to N$ that is the identity (the map $r$ restricted to $N$ is identity on $N$). Must the ...
0
votes
0answers
37 views

Hyper $n-$ torus cohomology group?

I don't know if this interpretation is correct. Is $S^{n_1} \times S^{n_2} \times \dots \times S^{n_k}$ some sort of hyper torus of dimension $1 + \sum_{k = 1} n_k$ (see here for calculation)? Let's ...
8
votes
1answer
254 views

Five lemma: unique isomorphism?

Consider the Five lemma with abelian groups. If $l$, $m$, $p$, and $q$ are isomorphisms, then $n$ is an isomorphism. Let $n'\colon C\to C'$ be a second homomorphism such that $ n' \circ g=s\circ m$ ...
2
votes
1answer
78 views

Question about two homeomorphic closed manifolds

I was studying about algebraic topology with my study group. So, there was a question that held all of the study members confused. If two closed manifolds are homeomorphic, they must have same ...
2
votes
1answer
132 views

A question about Hatcher exercise 2.1.23

I'm trying to solve a problem on barycentric subdivision. The problem deals with any delta complex in general, so I can't find a way to formulate some argument at all...I can't even see how to express ...
3
votes
1answer
105 views

How to picture a projective variety?

The picture of $\{(x:y:z) \in \mathbb P_{\mathbb C}^2 | yz =0\}$ is two spheres (each representing a copy of $\mathbb P_{\mathbb C}^1$) intersecting at one point (representing $(1,0,0)$). But ...
1
vote
1answer
97 views

How do I prove that there is no map $f:B^2\rightarrow S^1$ such that $f(-x)=-f(x)$ on $S^1$?

How do I prove that there is no map $f:B^2\rightarrow S^1$ such that $f(-x)=-f(x)$ on $S^1$? I'm not familiar with this kind of problems. I'm only comfortable with algebraic relations between ...
2
votes
1answer
100 views

How to pronounce Ext

Maybe this is a dull question, but I'm curious about how people pronounce the word 'Ext', for the $\operatorname{Ext}^{n}_{R}$ functor; some people called it as 'ee-ex-tee', 'eksit', or even just an $...
1
vote
1answer
50 views

If $f\circ q,g\circ q$ are homotopic on $S^1$, are $f,g$ homotopic?

Let $f,g:S^1\rightarrow S^1$ be continuous functions. Define $\alpha:[0,1]\rightarrow S^1:t\mapsto (\cos 2\pi t, \sin 2\pi t)$. If $f\circ \alpha$ and $g\circ \alpha$ are homotopic, then are $f,g$ ...
6
votes
0answers
176 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
4
votes
1answer
95 views

Visualising this CW structure for the $S^3$

I'm asked to prove that the following is a CW structure for the 3-sphere, (as a part of an exercise involving defining the Cw structure of the Lens Spaces) I'm asked to prove that the following is a ...
1
vote
1answer
78 views

How do I show that this map is path homotopic to a constant map?

Let $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ be null-homotopic loop. Since $\mathbb{C}\setminus\{0\}$ is path connected, $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ is homotopic to ...
6
votes
1answer
188 views

Atiyah K theory

On page 3 of Atiyah's book on K theory (link here: http://www.cimat.mx/~luis/seminarios/Teoria-K/Atiyah_K_theory_Advanced.pdf) he states: "Since a vector bundle is locally trivial, any section of a ...
2
votes
1answer
62 views

Combining homotopies

I have two homotopies $H,G:D^n \times I \to Z$ with $H(x,0) = f(x)$, $H(x,1) = f'(x)$, $G(x,0) = f'(x)$, $G(x,1) = g(x)$ for some maps $f,f',g:D^n \to Z$. $G$ has the additionnal property of being ...
1
vote
0answers
62 views

homotopy invariance for singular homology for maps of pairs

Let R be a ring, $(X,A), (Y,B)$ pairs of topological spaces and $f,g:(X,A)\to (Y,B)$ continuous maps of pairs such that there exists $H:X\times I\to Y$ homotopy with $H(A\times I)\subseteq B$, $H(x,0)=...
2
votes
0answers
57 views

Explicit expression for the topological invariant of O(n)

I learned that the fundamental group of $O(n)$ is $\Bbb{Z}/2\Bbb{Z}$ (for $n>2$). What is the explicit expression for its topological invariant? To be specific: Given a smooth path $\{M(t):T^1\...
3
votes
3answers
147 views

Homology and Homotopy in the Plane

Suppose we're living in the plane minus (possibly infinitely many) isolated points, which I'll call poles. Intuitively, the following two statements seem reasonable: Loops in the plane are homotopic ...
1
vote
1answer
100 views

simplicial homology in Hatcher book.

I was studying simplicial homology in Hatcher's Algebraic topology book.In one paragraph book says following: Some obvious general questions arise: Are the groups $H_n(X)$ independent of the choice of ...
0
votes
1answer
32 views

To what scope polar coordinate makes sense?

In basic calculus, one partial-differentiate a differentiable function whose domain is an open set or a closed set etc. However how formally this process works? Here is a reference : definition of ...
3
votes
1answer
125 views

Is this “the winding number”?

Note that $p:\mathbb{C}\rightarrow \mathbb{C}\setminus\{0\}:z\mapsto e^z$ is a covering map. Let $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ be a closed curve. Let $\gamma$ be any ...
2
votes
1answer
60 views

cohomology monomorphism between grassmannians and product of projective spaces

Let $S^1\times\cdots \times S^1( n\text{ times })=\prod_n U(1)\to U(n)$ be the inclusion. This induces a map between classifying spaces $$ \prod_nBS^1\to BU(n).$$ i.e., $$ (\mathbb{C}P^\infty)^{\...
1
vote
2answers
42 views

How do I prove $e^z$ is a covering map using this fact?

I have proven that $p:\mathbb{R}\rightarrow S^1:t\mapsto (\cos 2\pi t,\sin 2\pi t)$ is a covering map and $S^1$ and $\mathbb{C}\setminus\{0\}$ are homotopically equivalent. Using these facts, how do ...
2
votes
1answer
57 views

Identifying the orbit space of the unitary group $U(n)$ in the compact symplectic group $Sp(n)$

Let $Sp(n)$ be the compact symplectic group. Let $U(n)$ the unitary group, and $O(n)$ the orthogonal group. What is $Sp(n)/U(n)$? What is $U(n)/O(n)$? I obtain that $Sp(1)/U(1)=S^3/U(1)=\mathbb{C}P^1\...
0
votes
1answer
72 views

Think of the surface of genus $k$ as a sphere with $k$ tubes sewn in. Calculate its Euler characteristic by trangulating.

Think of the surface of genus $k$ as a sphere with $k$ tubes sewn in. Calculate its Euler characteristic by trangulating. I know that I need to make the genus covered by infinitely many triangle then ...
6
votes
1answer
225 views

How was real analysis & topology taught in the 70's?

What was the 'gold standard' textbook before Rudin? Furthermore, if anyone has knowledge of what textbooks Princeton or Harvard used back in the 1960's or 70's, I would highly appreciate it if you ...
1
vote
0answers
13 views

How do I show that brouwer's theorem holds for this domain?

Define $\alpha:[0,1]\rightarrow \mathbb{R}^2:t\mapsto (\cos 2\pi t, \sin 2\pi t)$. Let $\gamma:[0,1]\rightarrow\mathbb{R}^2$ be a loop at $(1,0)$ homotopic with $\alpha$. Let $D$ be the inside ...
2
votes
1answer
67 views

What is the Cayley graph of $(\Bbb Z/2\Bbb Z)\times(\Bbb Z/2\Bbb Z)$?

I get that the presentation of the new group, with respect to two generators, would be $(x,y \;|\; x^2= y^2=1)$ but I'm not sure how the actual graph would look. Would it consist of an infinite ...
1
vote
1answer
201 views

Show that the free group on $n$ generators is a finite index subgroup of $F_2$

Using covering spaces, prove that for each integer $n \geq 2$, $F_n$ is a finite index subgroup of $F_2$, where $F_n$ is the free group on $n$ generators. I get how the cayley graph of $F_n$ would be ...
3
votes
1answer
59 views

Based covering maps for a bouquet of two circles

For each of the following subgroups of $$ \left \langle x,y \right \rangle = \pi_{1}(S^{1}\vee S^{1}) $$ construct a based covering map $$ \ p:(\tilde{X},\tilde{b})\rightarrow (S^{1}\vee S^{1},b) $$...
2
votes
2answers
123 views

wedge product of projective planes

if we have the wedge product of the real projective plane $P^2$ V $P^2$ Then how would i use Seifert Van Kampens theorem to compute the fundamental group $\pi_1$($P^2$ V $P^2$ ) ? i'm some what ...
1
vote
1answer
52 views

Answer gap-filling-in topology, describing the kernel from the Seifert–van Kampen theorem

The question is: Let $X=S^1\times I$ and let $A=S^1\times[0,3/4)$ and $B=S^1\times(1/4,1]$ So that $\{A,B\}$ is an open cover. I have been tasked with using the the Seifert-van Kampen theorem to ...
2
votes
0answers
111 views

When n is odd, an even map $S^n\rightarrow S^n$ always has even degree.

If $f$ is an even map $S^n$ to $S^n$ then this induces an map $S^n$ to $RP^n$ to $ S^n$ Also when n is odd we have that $H_n(RP^n)$ is isomorphic to $H_n(RP^n/RP^{n-1})$. I would like to use this to ...
1
vote
1answer
61 views

CW Structure of $SU$

I'm reading Switzer's Algebraic Topology and he mentions that $SU = SU(\infty)$ can be given a CW complex structure. He also says that this implies, by a theorem of Milnor's, that $\Omega SU$ has the ...
0
votes
1answer
180 views

De Rham cohomology of $\mathbb{R}^2 \setminus \{k~\text{points}\}$

This question is motivated by Exercise 1.7 from Differential Forms in Algebraic Topology by Bott & Tu. The original question in the text concerns the de Rham cohomology of $\mathbb{R}^2$ with ...
3
votes
1answer
98 views

fibration of real projective space over complex projective space

From the fibration $$U(1)=S^1\to S^{2n+1}\to \mathbb{C}P^n,$$ can we quotient the action of $S^0$ and obtain a well-defined fibration $$ S^1/S^0=\mathbb{R}P^1\cong S^1\to \mathbb{R}P^{2n+1}\to\mathbb{...
2
votes
1answer
88 views

fibration of complex projective space over quaternion projective space

From the fibration $$Sp(1)=S^3\to S^{4n+3}\to \mathbb{H}P^n,$$ can we quotient the action of $U(1)=S^1$ and obtain a well-defined fibration $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
1
vote
0answers
62 views

Is there a general notion of orientability, e.g. for the rationals?

I was discussing orientability with a friend today. To me, orientation is a subtle concept I hardly understand. To get my perspective across, I was trying to come up with spaces which are intuitively ...
3
votes
1answer
83 views

Postnikov towers for non-CW spaces

In the literature, Postnikov systems seem to be defined always in the setting of CW complexes. Looking at the proofs, it is not clear to me, why this assumption should be necessary. Question: Does ...
1
vote
1answer
89 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow \mathbb{C}P^\infty,...
3
votes
1answer
122 views

Can every basic concept of fundamental group be generalized to homotopy group?

I'm taking (undergraduate) algebraic topology this year and I have learned some basic concepts in this subject. I found this subject interesting, but it seems like the usefulness of fundamental groups ...
5
votes
1answer
74 views

$H_2(M)$ is free abelian for any simply connected $4$-manifold

In Naber's book "Topology, Geometry and Gauge Fields. Foundations", it is stated that for each $4$-manifold $M$ which is smooth, closed, connected and simply connected we have $H_0(M) = H_4(M)= \...
3
votes
1answer
136 views

How to prove that $CP^4$ cannot be immersed in $R^{11}$

Please let me know how to prove that $CP^4$ cannot be immersed in $R^{11}$. I know a proof using an integrality theorem for differentiable manifolds but I want to know if a more direct and simple ...