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

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Fundamental group and Fuchsian group

We know that the fundamental group of a compact surface of genus larger or equal to 2 is a Fuchsian group, i.e. a discrete subgroup of the automorphism group of the hyperbolic plane PSL(2,R). And any ...
3
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1answer
93 views

Non homeomorphism

I want to show that the sphere $S^2$ and the torus $T^2$ are not homeomorphic, using the notion of intersection modulo $2$. I have to show that any two loops on the sphere $S^2$ have an even number of ...
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1answer
80 views

Covering space action on an orientable manifold $M$ implies $M/G$ orientable (Hatcher)

I'm trying to solve the following problem from Hatcher (3.3.4) Given a covering space action of a group $G$ on an orientable manifold $M$ by orientation preserving homeomorphisms, show that $M/G$ is ...
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0answers
76 views

Understanding the “least normal subgroup” in Seifert van Kampen

The Seifert van Kampen theorem implies that if $V$ is simply connected then there is an isomorphism $k: \pi_1(U, x_0) / N \rightarrow \pi(X, x_0)$ where $N$ is the least normal subgroup of $\pi_1(U, ...
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1answer
59 views

Proving a fundamental group is NOT abelian

I was wondering if the following approach would be possible in proving the fundamental group of $X$ was not abelian. If one can show there exists a homomorphism: $\pi_1(X, x_0) \rightarrow ...
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1answer
55 views

Manifold with special cohomology group

I am trying to find if there is a orientable compact manifold $M$ of dimension 10 with the 5th cohomology group of De Rham $H^5_{DR}(M)\cong \mathbb R$. But, I can find such an example or prove that ...
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2answers
79 views

Show $H_2(M, \mathbb{Z}) = \mathbb{Z^r}$ if $M$ is orientable, $\mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$ if nonorientable

I'm trying to solve this problem from Hatcher 3.3.24. Let $M$ be a closed connected 3-manifold, and write $H_1(M, \mathbb{Z})$ as $\mathbb{Z^r} \oplus T$ where $T$ is torsion. Show that $H_2(M, ...
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1answer
94 views

Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even

I want to prove: Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even. I'm stuck. My thoughts are: I want to use the hairy ball theorem and I want to ...
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2answers
138 views

Fundamental group a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane.

Find the fundamental group of the space comprising a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane. Touching means having one point in common. I ...
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0answers
35 views

Alexander Method for non-orientable Surface

From Farb and Margalit,Primer on Mapping Class Group, p.62, we know For example for torus with four puncture we can choose following claret red curves. Curves are choosen such that their complement ...
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1answer
113 views

Converse to the Eilenberg-Steenrod theorem?

For the purposes of this question, a homology theory is a covariant functor from the homotopy category of finite pointed CW complexes to graded abelian groups, and a collection of connecting ...
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1answer
28 views

Fundemental group of $D^2\setminus\{x\}$

Let $D^2=\{x\in\mathbb{R}^2:||x||\le1\}$, $x\neq a\in D^2$. Find $\pi_1(X\setminus\{x\},a)$ if: a. $x\in\partial D^2$ b.$x\in \text{int} D^2$ about the first one I think the fundamental ...
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3answers
73 views

Complete metric space, not simply-connected

I've been going over the algebraic topology part of Munkres and this question has stumped me. If we have a complete metric space that is not compact, must it be simply-connected (path-connected plus ...
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0answers
42 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
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1answer
78 views

Counterexamples to Brouwer’s fixed point theorem

Brouwer’s fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least ones fixed point. If we replace the convex condition with let's say ...
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1answer
68 views

Show that for a degree 1 map $f: M \rightarrow N$ the induced map $f_*: H_1(M) \rightarrow H_1(N)$ is a surjection

I'm trying to solve the following problem: Show that for a degree 1 map $f: M \rightarrow N$ of connected, closed, orientable manifolds, the induced map $f_*: \pi_1(M) \rightarrow \pi_1(N)$ is ...
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2answers
233 views

Does the projective Fermat curve have singular points?

I want to check if the projective Fermat curve, $$X^n+Y^n+Z^n=0, n \geq 1$$ has singular points. Could you give me a hint how we could do this? Do we have to check maybe if the curve is ...
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1answer
124 views

Reference Request: Thom Spectrum of a virtual vector bundle

Given a vector bundle $\xi$, one can construct a Thom spectrum $\mathbb{T}h(-\xi)$ associated to its additive inverse. I unsuccessfully browsed through literature to find a reference for an explicit ...
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4answers
124 views

Is a continuous bijection function from a hausdorff space to a compact space is a homeomorphism?

We know a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism. But I am wondering what happened if we switch the domain and codomain. Is a continuous bijection ...
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1answer
217 views

Is the complex projective plane a compact manifold with or without boundary (closed manifold)?

my question is the one in the title. (My motivation is to understand in which way Freedman's classification of compact simply-connected 4-manifolds implies the Poincare conjecture for 4-manifolds, as ...
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0answers
49 views

Finish the proof of Borsuk-Ulam theorem (Hatcher)

Hatcher at page 229 proposes to prove the Borsuk-Ulam theorem using the fact that any continuous map $f \colon\mathbb R P^n \to \mathbb RP^m$, $n > m$, induces the trivial map in cohomology with ...
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2answers
145 views

How to show the covering space of an orientable manifold is orientable

I'm trying to prove this using purely topological arguments, no differential geometry as I haven't been exposed to it. I've been playing around with definitions a bit and here's what I have so far. ...
3
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1answer
88 views

Original Papers on Singular Homology/Cohomology.

I am currently reading Singular Homology Theory and Cohomology on my own mainly from Hatcher's "Algebriac Topology" and "Topology and Geometry" by Bredon. Quite often it happens that it takes a lot of ...
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1answer
113 views

Manifolds with vanishing Stiefel-Whitney classes but are not stably parallelizable

It is known that if a manifold is stably parallelizable, then it's Stiefel-Whitney classes must vanish. Is the converse true? Note that we know that the converse cannot hold if stably parallelizable ...
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3answers
255 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
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2answers
78 views

Is $\text{Hom}(A\oplus B, G) = \text{Hom}(A, G)\oplus \text{Hom}(B, G)$ true?

I'm reading Hatcher's Algebraic Topology, and in the proof of the Universal Coefficients Theorem (Page 192), it says for abelian groups $A$ and $B$, and an arbitrary group $G$, we have ...
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0answers
57 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 $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
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1answer
43 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^*: ...
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1answer
72 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 ...
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1answer
92 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 ...
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0answers
84 views

About mapping cone complex

Let $X$ be a topological space. Define two cochain complexes $\mathcal{C}$ and $\mathcal{D}$ by $\mathcal{C}=\{C^k(X; \mathbb{Q}), \partial^k\}, \qquad\mathcal{D}=\{C^k(X; \mathbb{R}), \partial^k\},$ ...
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1answer
34 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
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0answers
59 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
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1answer
83 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
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0answers
33 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 ...
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3answers
287 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
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1answer
67 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
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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 ...
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1answer
29 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 ...
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0answers
22 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 ...
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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 $N \cong S^n \times S^n$ ...
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5answers
105 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 ...
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2answers
77 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 ...
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1answer
34 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 ...
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0answers
34 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 ...
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1answer
209 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
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1answer
58 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 ...
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54 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
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1answer
84 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 ...
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1answer
86 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 ...