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

learn more… | top users | synonyms (1)

0
votes
0answers
28 views

cohomology homomorphism between grassmannians induced by inclusion

Let $i:G_k(\mathbb{R}^n)\to G_k(\mathbb{R}^\infty)$ be inclusion of grassmannians. Then $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
1
vote
1answer
34 views

cohomology homomorphism induced by classifying map

Let $\xi=(E,p,B)$ be an $n$-dimensional vector bundle. Let $f: B\to G_n(\mathbb{R}^\infty)$ be the classifying map. Let $f^*: ...
2
votes
1answer
53 views

Classical proof of Brouwer fixed point theorem: why the projection is continuous?

In the classical proof of the Brouwer fixed point theorem, we suppose for absurd that if $f$ is a continuous function from the closed unity ball to itself with no fixed point, then we show that it's ...
2
votes
1answer
69 views

Topological Space with Given Fundamental Group

We know that if we want to construct a space with a given fundamental group $G$ ,we can use cells and attaching maps, or fundamental domains and attaching maps, as in : How to determine space with a ...
1
vote
0answers
17 views

About mapping cone complex

Let $X$ be a topological space. Define two cochain complexes $\mathcal{C}$ and $\mathcal{D}$ by $\mathcal{C}=\{C^k(X; \mathbb{Q}), \partial^k\}, \qquad\mathcal{D}=\{C^k(X; \mathbb{R}), \partial^k\},$ ...
1
vote
1answer
26 views

Restricted join operation on the simplicial complex?

Let $A$ and $B$ be two simplicial complexes (or CW-complexes) containing a common subcomplex C. Assume that C is contractible in both A and B. Is it true that the space obtained gluing A and B over ...
2
votes
0answers
30 views

cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...
3
votes
1answer
73 views

Cancellation of Direct Product in Top

I'm thinking to the famous problem of cancellation property in Top, i.e: $$T_1 \times T_2 \cong T_1 \times T_3 \Rightarrow T_2 \cong T_3. $$ Clearly there are many counterexamples like $\prod_{i \in ...
2
votes
0answers
21 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 ...
6
votes
3answers
188 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
38 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
55 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 ...
0
votes
1answer
18 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
0answers
17 views

Configuration space of product spaces

Let $M,N$ be manifolds. Let $F(M,n)$, $F(N,n)$ be ordered configuration spaces of order $n$. Let $F(M,n)/\Sigma_n$, $F(N,n)/\Sigma_n$ be the unordered configuration spaces of order $n$, for ...
0
votes
1answer
28 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 $N \cong S^n \times S^n$ ...
5
votes
5answers
60 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
57 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$ ...
2
votes
1answer
46 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 ...
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
56 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
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
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 ...
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 $f,g$ ...
6
votes
0answers
132 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 ...
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 ...
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 ...
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 ...
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 ...
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 $H(A\times I)\subseteq B$, ...
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 ...
3
votes
3answers
69 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
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 ...
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 ...
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., $$ ...
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 ...
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
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 ...
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
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 ...
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 ...
3
votes
1answer
22 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
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
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 ...