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

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3
votes
2answers
58 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
31 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
25 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 ...
7
votes
1answer
176 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$ ...
1
vote
1answer
76 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 ...
3
votes
2answers
114 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
3
votes
1answer
58 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 ...
0
votes
0answers
31 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 ...
3
votes
1answer
63 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
26 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 $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
3
votes
1answer
69 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 ...
6
votes
0answers
134 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 ...
0
votes
0answers
23 views

maps between suspension of complex projective spaces and special unitary groups

How to do the following question? I get totally lost... this question is given by the professor in our final exam paper.
1
vote
0answers
24 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 $f,g$ ...
1
vote
1answer
59 views

How to visualize topological differences between $\mathbb{R}P^{2n}$ and $\mathbb{C}P^n$

I never stopped to really understand the topology of $\mathbb{R}P^{2n}$ and $\mathbb{C}P^n$. In my Algebraic Topology class, we calculated the homology of them, and they are significantly different. I ...
2
votes
1answer
47 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 ...
5
votes
4answers
743 views

Homotopy composition Hatcher exercise

Show that composition of paths satisfies the following cancellation property: if $f_0 \cdot g_0 \simeq f_1 \cdot g_1 $ and $g_0 \simeq g_1$, then $f_0 \simeq f_1$. So I have two homotopies. So say ...
0
votes
1answer
22 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 ...
0
votes
1answer
51 views

Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows ...
2
votes
0answers
51 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 ...
1
vote
1answer
28 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 ...
1
vote
0answers
38 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 $H(A\times I)\subseteq B$, ...
1
vote
1answer
50 views

is configuration space an H-space?

Let $X$ be a manifold. Let $F(X,n)$ be the configuration space of order $n$. Let $B(X,n)=F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. Is $B(X,n)$ an H-space? Under what ...
0
votes
1answer
46 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
131 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 ...
14
votes
1answer
314 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
0
votes
1answer
68 views

How do I show that $\mathbb{R}^n$ is simply-connected?

I have shown that $\mathbb{R}$ is simply connected by reparametrization. However, how do I show that $\mathbb{R}^n$ is general?
1
vote
1answer
30 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
24 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 ...
2
votes
1answer
33 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., $$ ...
3
votes
1answer
48 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 ...
7
votes
2answers
174 views

Understanding Hatcher's proof of Borsuk-Ulam theorem for $n=2$

I am trying to understand the proof of the Borsuk-Ulam theorem for $S^2$ given in Hatcher's "Algebraic Topology" (Th. 1.10), as another person does here, but we are stuck at different places, so I ...
4
votes
2answers
158 views

A Question about Borsuk-Ulam Theorem

I don't understand a step of Borsuk-Ulam theorem, which i tagged with a star below. $\underline{Borsuk-Ulam}$: If $f:S^2\rightarrow\mathbb R^2$ continuous, then $\exists x$, s.t. $f(x)=f(-x)$ ...
2
votes
1answer
35 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 ...
0
votes
2answers
22 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 ...
1
vote
1answer
35 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
0answers
9 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 ...
1
vote
1answer
27 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 ...
1
vote
1answer
35 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 ...
2
votes
2answers
39 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
28 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
1answer
41 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 ...
3
votes
1answer
68 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 ...
2
votes
0answers
43 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 ...
3
votes
1answer
48 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
0answers
47 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 ...
0
votes
0answers
37 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 ...
0
votes
0answers
23 views

all differentials collapse of the Serre spectral sequence

Let fibration $$ SO(n)\to SO(n+1)\to S^n, $$ consider the Serre spectral sequence of cohomology $(E^{*,*}_k,d_k)$, $k\geq 2$, $E^{p,q}_2=H^p(S^n;\mathbb{Z}_2)\otimes H^q(SO(n);\mathbb{Z}_2)$. How ...
6
votes
1answer
162 views

What is a good, hi-tech textbook on complex analysis?

I am looking for an introductory textbook for Complex Analysis that is hi-tech. All the books I have looked at suffer from the same problem; they're only assuming that the reader is familiar with is ...
1
vote
1answer
35 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 ...