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

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23 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
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0answers
28 views

Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
4
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1answer
24 views

Completing compact surfaces with boundary to closed surfaces in $\mathbb R^3$

My question is whether any compact smooth surface in $\mathbb R^3$ (with smooth boundary) can be completed to a closed smooth surface in $\mathbb R^3$ without boundary? It is easy to complete it to an ...
3
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1answer
17 views

how to compute the de Rham cohomology with compact support of a mobius strip

I am having problem computing the de Rham cohomology with compact support of an open mobius strip,it's aquestion from Bott's book, and Bott said its cohomology is identically zero which can be ...
5
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2answers
66 views

Finding the de Rham cohomology of an open subset of $ \Bbb{R}^{n} $ minus a point.

Here’s my question: Let $ n \in \mathbb{N}_{\geq 2} $. Suppose that $ U \subseteq \Bbb{R}^{n} $ is an open set and that $ x \in U $. Then show that $$ {H_{\text{dR}}^{n - 1}}(U \setminus \{ x ...
2
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1answer
31 views

Join of closed embeddings is a closed embedding

An exercise from James's book General Topology and Homotopy Theory asks the reader to prove that if $\phi_1:X_1 \to Y_1$ and $\phi_2:X_2 \to Y_2$ are closed topological embeddings, then $\phi_1 * ...
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0answers
45 views

Cohomology ring of $S^3 \setminus A $ and $S^3 \setminus B $,where $A$ is union of two once linked circle and $B$ is union of two unlinked circles

Suppose $A$ is union of two once linked circles in $S^3 $ and $B $ is union of two unlinked circles.show that $S^3 \setminus A $ and $S^3 \setminus B$ have same cohomology group but not same ...
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0answers
47 views

Cup product Structure of $X \vee Y $

Suppose $\alpha \in H^*(X)$ and $\beta \in H^*(Y)$ are of positive degrees. Show that $\alpha\beta=0$ in $H^*(X \vee Y)$. I am unable to show that. I think $\alpha\beta=0 $ because intersection of ...
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0answers
13 views

lemma( lifting- cantor path )? [on hold]

Definition: Let $p:E\rightarrow B$ be a map. If $f:X\rightarrow B$is a map, a lifting of is a map $\widetilde{f}:X\rightarrow E$ such that $p\circ \widetilde{f}=f$ ¿TRUE or FALSE? "Let $C$= Cantor ...
1
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1answer
38 views

What exactly is the Kahler class of a torus?

Is there an intuitional way to understand what the Kahler class of $T^2$ actually is? It would be extremely useful to me if you could provide me some intuition behind it!
2
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2answers
29 views

Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
1
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1answer
39 views

computing Lefschetz number

We have a fixed point theorem which says that : Let $X$ be a compact polyhedron, $f:X\rightarrow X$ be a continuous map. If $L(f)\neq 0$ then $f$ must have a fixed point. (Lefschetz number is ...
4
votes
1answer
68 views

If $f:S^1\to S^1$ doesn't have any fixed point then it is homotopic to the identity

How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
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0answers
25 views

thorough proof of Mayer-Vietoris implies Excision

Where could I find a very complete proof of how the Mayer-Vietoris sequence implies the Excision theorem? I've read a few proofs, but they always leave out the details! Thank you!
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0answers
26 views

Curve concatenation in manifolds.

I am having difficulty understanding what is going on geometrically when you add together multiples of curves (1-chains) in a differentiable manifold. Say we have two curves $A$ and $B$ together with ...
2
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3answers
44 views

simply connected covering of a path connected space (II)

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
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0answers
38 views

Fundamental group of the mapping cone of a loop

You can read in Wikipedia the following: Given a space X and a loop $\alpha\colon S^1 \to X$ representing an element of the fundamental group of $X$, we can form the mapping cone $C_α$. The effect of ...
3
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1answer
35 views

Calculating the homology groups of a simplicial complex using a Mayer-Vietoris sequence

I'm trying to calculate the homology groups for a simplicial complex $X$, which is a union of subcomplexes $X_1$ and $X_2$ which are both combinatorially equivalent to cones. This is the information I ...
2
votes
0answers
21 views

Stiefel-Whitney Classes of a submanifold

Suppose we have a manifold $M$ that is the product of $k$ copies of real projective space, say $$ M = \prod_{i=1}^k \mathbb{R} P^{n_i}.$$ Then $H^*(M; \mathbb{Z}/2) = \mathbb{Z}/2 [ \alpha_1, \dots, ...
6
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1answer
86 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...
3
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2answers
79 views

Finding the Fundamental Groups of Some Modular Spaces

I'm looking to compute the fundamental group of a couple of different quotients of the $n$-torus. The first of these I'm interested is the space $\mathbb{T}^n/S_n$ where the symmetric group $S_n$ ...
3
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1answer
57 views

Understanding Hatcher's proof for $\chi(M)=0$ for non-orientable manifolds $M$ of odd dimension

In the Corollary 3.37 Hatcher proves that for a closed odd-dimensional manifold $M$, its Euler characteristic is zero. The first part of the proof deals with orientable manifolds, and uses Poincare ...
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2answers
30 views

Covering maps question [on hold]

I need to show that, given $p_1:Y_1\rightarrow X $ and $p_2:Y_2\rightarrow X $ two covering maps locally pathwise connected, and there exists $\alpha$ a covering homomorphism between p1 and p2, then ...
2
votes
1answer
55 views

Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...
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0answers
29 views

Betti numbers over unital rings [on hold]

Is the following statement correct? Given a manifold $M$. If $H_1(M,\mathbb Z)$ is a finite cyclic group, then the first $R$-Betti number $b_1(M,R)$ is bounded from above by $1$ for every unital ring ...
1
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1answer
37 views

Injection of the mapping cone of $z^2$

We define the mapping cone of $f:S^1\to S^1=:Y$, $f (z)=z^2$ as the quotient space of $S^1\times [0,1]\sqcup Y$ where $(z,0)$ and $(z',0)$ are identified and where $(z,1)$ and $f(z)$ are identified ...
1
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1answer
39 views

Write down a map $f$ from the torus $T$ to itself such that the induced map $g:H_1(T) \to H_1(T)$ is given by the matrix ( 1 1 : 0 1)

I think $f(x,y)=(x,x+y)$. suppose $f(x,y)=(x,x+y)$.then I am looking at the action of $g$ on the generators of $H_1(T)$. but I can't show that.
1
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1answer
32 views

Quotient of union of two spaces

Let $X$ be a topological space, $f : S^{n-1} \to X$ and $Y := X \cup_f D^n = \big(X \coprod D^n\big) / \sim$ , where $t \sim f(t)$ for $t \in S^{n-1}$. Problem. Prove that $Y/X \cong S^n$. My idea. ...
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0answers
37 views

What do you get if you glue a disk twice around a circle?

I would like to know what you get if you glue the disk $D^2$ around the circle $S^1$ via the map $\phi \colon \partial D^2\to S^1$, $\phi (e^{i\theta})=e^{2i\theta}$. I would have thought you would ...
1
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1answer
70 views

Cohomology Group of $CP^2 \wedge CP^2$

Calculate the cohomology group of $CP^2 \wedge CP^2$ To do this, at first I am trying to calculate the homology group and then use Universal Coefficient Theorem. To do this, at first I have ...
1
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1answer
49 views

What is the difference between CW-complex and Cellular complex?

Is every CW-complex is a Cellular space? Is its converse true? If it is true then what is the difference between them? We include the definition of CW-complex in algebraic topology given by ...
2
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1answer
53 views

Equivalent presentation for the fundamental group of the projective plane

We know that $\langle a,b;(ab)^2=1\rangle$ and $\langle z;z^2\rangle$ are presentations of the fundamental group of the projective plane. Therefore, one is obtained from the other via Tietze ...
3
votes
1answer
56 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
4
votes
1answer
63 views

Milnor's definition of bundle map in “Characteristic Classes”

In chapter 3 of "Characteristic Classes", Milnor defines bundle maps, requiring them to map fibers isomorphically onto fibers. Why not merely require homorphisms on fibers? (e.g., for the given ...
5
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0answers
45 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
0
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0answers
30 views

An example of $K(G,1)$ in Hatcher

A $K(G,1)$ space is a path-connected topological space $X$ with contractible universal cover and $$ \pi_1(X)=G. $$ I am reading about $K(G,1)$ spaces in Hatcher's textbook and I don't understand ...
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0answers
23 views

Immersion of punctured torus into Euclidean [duplicate]

(a) Show there is an immersion of the punctured torus $S^1\times S^1$ - {a point} into $R^2$. (b) generalized it to $T^n$ - {a point} into $R^n$ can you give concrete proof for these problem? ...
1
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1answer
53 views

Euler characteristic of a singular fiber

I am trying to understand Kodaira's classification of fibers. In the table at page 41 of Miranda's book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf there is given the Euler number of the ...
0
votes
1answer
28 views

Simple homotopy construction

I'm sure this isn't too difficult but i can't seem to do it if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop How would i construct an explicit homotopy between the ...
2
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0answers
28 views

Section of a covering projection from a connected space [duplicate]

Let $p:\overline{X}\rightarrow X$ is a continuous mapping. A continuous map $s:X\rightarrow \overline{X}$ such that $p\circ s =Id_X$ is called a section of $p$. Suppose $\overline{X}$ is connected ...
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1answer
21 views

The fundamental group of some wedge sum

I was wondering how one can compute the fundamental group of the wedge sum of a sphere and 2 circles , i know the fundamental group is Z*Z ,and that the fundamental group of a wedge sum is the free ...
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0answers
19 views

Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
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1answer
39 views

Show that $p:SO_3 \to\mathbb S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle

Show that $p:SO_3 \to\mathbb S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle. I know that $SO_3$ acts on $\mathbb S^2$ transitively saying that $p$ is onto.I have a problem with local ...
1
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1answer
43 views

Homology of a simplicial set

Let $X$ be a simplicial set. Define the complex $(C^X_\bullet,D)$ by $$C^X_n=\bigoplus_{X_n} \mathbb{Z}$$ and $$D_n=\sum_{i=0}^n (-1)^i d_i:C_n \to C_{n-1}$$ where the $d_i$'s are the face maps. I ...
3
votes
1answer
32 views

Embeddability of connected sum of non-embeddable surfaces

Let $X$ be a surface which can not be embedded into $\Bbb R^n$. Let $X \# X $ denotes the connected sum of two copies of $X$. Then is it true that the connected sum $ X \# X $ is also not embeddable ...
2
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1answer
84 views

Examples with zero first Stiefel-Whitney class and nonzero second Stiefel-Whitney class

What's the simplest/most concrete vector bundle you can think of that has zero first Stiefel-Whitney class but non-zero second? That would be the simplest space that doesn't have spinors. (See Spin ...
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0answers
15 views

Morphism of modules of sections of pullback bundles

Suppose that we have a morphism $\theta: \Gamma(B,E_1) \to \Gamma(B,E_2)$ where $E_i$ are two vector bundles over $B$ and let $f:A \to B$ be a continuos map. Then we can define a pullback bundles ...
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1answer
10 views

Section of pullback bundle

Suppose that $E \to B$ is a vector bundle and $f:A \to B$ is continuos. If $s$ is a section of $E$ how to define a section of pullback bundle? On wikipedia they say that it induces the section of the ...
2
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2answers
64 views

A problem on covering space from Hatcher book…

I was trying a problem from Hatcher's book Algebraic Topology, in section 1.3 problem number 12. Let $a$ and $b$ be the generators of $\pi_1(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. ...
1
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2answers
54 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...