Tagged Questions

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

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-1
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0answers
14 views

Show that the quotient map G-> G/H is a covering space [duplicate]

G be a topological group.H be a subgroup of H.suppose that the subspace topology on H is the discrete topology.Show that the quotient map G-> G/H is a covering space. Prove that the quotient map $P:G ...
1
vote
0answers
11 views

The smallest $n> 0$ with the nonzero $n$th stiefel whitney class is a power of 2 when total stiefel whitney class is not trivial.

This is the Problem 8-B form the characteristic classes by John W. Milnor and James D. Stasheff. [Problem 8-B] If the total stiefel whitney class $w(\xi) \neq 1$, show that the smallest $n>0$ with ...
0
votes
1answer
24 views

algebra of polynomials of non-commuting variables

In Hatcher's book algebraic topology page 496-497: the Steenrod algebra $\mathcal{A}_2$ is the algebra over $\mathbb{Z}_2$ formed by polynomials of non-commuting variables $Sq^1,Sq^2,\cdots$ modulo ...
4
votes
1answer
37 views

Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?
0
votes
0answers
11 views

Explicit example of $\Delta$-complex of the Torus

In "Algebraic Topology"-Hatcher the beginning of the chapter on $\Delta$-complexes shows a pictorial example of the $\Delta$-complexes of the Torus, $\mathbb{R}P^2$ and the Klein Bottle. In a ...
1
vote
1answer
43 views

Let $p: E \to B$ be a covering map. If $B$ is a completely regular space then prove that (edited) $E$ is completely regular space.

Let $p: E \to B$ be a covering map. If $B$ is a completely regular space then prove that $E$ is completely regular space. I am getting no clue how to construct the function $f$. The readers may ...
1
vote
1answer
35 views

How does one actually take the dual of a chain complex?

I know the following about the chain complex used for computing the homology groups of the torus $S^1 \times S^1$: The complex is $0 \to^{\delta_3} \mathbb{Z}[U] \oplus \mathbb{Z}[L] \to^{\delta_2} ...
2
votes
1answer
23 views

Euler characteristic of closed surface

Assume that you have a closed surface that can be covered by finitely many triangles. Then $K(p)= 6-val(P)$ where P is a vertex and $val(P)$ the number of edges that lead to this vertex. Now, I am ...
1
vote
0answers
24 views

Cones over a topological space and homotopy!

Let $X$ be a topological space and let $CX$ be the cone over $X$. We identify $X$ with the subspace $X\times \{0\}$ of the cone through the immersion $x \mapsto [(x,0)]$ for $x \in X$. Let $f:X \to Y$ ...
0
votes
0answers
18 views

smash product does not preserve equivalences: non-example

If $f : X \rightarrow Y$ is a weak equivalence of pointed spaces and $Z$ is a pointed space, then $f\wedge Z : X\wedge Z \rightarrow Y\wedge Z$ is not necessarily a weak equivalence, unless we're ...
0
votes
1answer
17 views

Fundamental Confusion Regarding the Fundamental Group

Consider the following question. Let $X$ be a topological space and $x_0$ be any point in $X$. Let $\gamma\in \pi_1(X,x_0)$ be a non-constant loop about the base point $x_0$. Then can it happen ...
0
votes
1answer
26 views

Algebraic Topology problem help! A mapping from the n-sphere to some set!

So let $f:S^{m}\to X$ be a continuous mapping. How can I prove that a) $f$ is homotopic with the constant mapping (i.e. the point I guess) and b) that $f$ can be augmented to a new mapping $f': ...
1
vote
0answers
31 views

Homotopy equivalence of pushouts of topological spaces

Let $h \colon A \to B$ and $r \colon S^{n-1} \to A$ be continuous maps. Assume that $h$ is an homotopy equivalence, prove that $$ D^n \cup_{r} A \simeq D^n \cup_{h \circ r} B$$ where $D^n ...
1
vote
2answers
15 views

Liftings of Nullhomotopic Maps

Is the lifting of a nullhomotopic map always a loop in a path connected space? This question is perhaps very trivial,but I am struggling with it. Please help.
1
vote
2answers
21 views

quotient topology in compact hausdorff space

Let X is compact hausdorff space and A is a subset of X. Show that X/A is Hausdorff ,when A is closed. I have no idea how to proof it. Can you give me a clue?
7
votes
3answers
128 views

Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
1
vote
0answers
15 views

Normal subgroup invariant under $\text{Ad}_g$

Denote by $G$ a Lie group with corresponding Lie algebra $\text{Lie}(G)$. There the three maps inner automorphism/conjugation: $\text{Int}_g = L_{g^{-1}} \circ R_g \in \text{Aut}(G)$, $\text{Ad}_g ...
2
votes
1answer
39 views

Extenting a homeomorphism on subsurface to the entire surface

Suppose you have surface and a subsurface. The complement of subsurface is union of open discs and once punctured open discs. Can all homeomorphisms of the subsurface be extended to homeomorphisms of ...
0
votes
0answers
11 views

What is $F_i^p(e_j)$?

$F_i^p$ is the face opposite the $i^{\text{th}}$ vertex in a $p$-simplex. My book mentions the term $F_i^p(e_j)$, where $e_j$ is a vertex in the simplex. What is $F_i^p(e_j)$?
1
vote
1answer
29 views

Homologous to zero but not contractible

Looking for instructive examples on the difference between homology and homotopy, I found here the following example: Example: Consider an oriented loop separating a genus $2$ surface into two ...
1
vote
0answers
19 views

Topology of operator bundle?

I am trying to understand the family version of the Atiyah-Singer Index Theorem as described in the book "Spin Geometry" by Lawson/Michelsohn. In Part III.§8, they define the operator bundle $$ ...
0
votes
0answers
30 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 ...
6
votes
1answer
51 views

If it looks like a solid torus, walks like a solid torus, and quacks like a solid torus, is it a solid torus?

If an orientable 3-manifold $M$ has boundary the torus $S^1\times S^1$ and deform retracts to a solid torus $S^1\times D^2$, is it necessarily homeomorphic to a solid torus? Equivalently, if the ...
1
vote
1answer
37 views

Is there a name for a function such that $f=e^g$?

Let $X$ be a topological space. Let $f:X\rightarrow \mathbb{C}\setminus\{0\}$ be a continuous function. Is there a terminology to call functions $f$ such that $f=e^g$ for some continuous map ...
0
votes
0answers
24 views

Are $z^n$ and $p(z)/|p(z)|$ homotopic?

Let $p:\mathbb C\longrightarrow \mathbb C$ be a complex polynomial with no zeros and degree $n$. Is it true that the maps $f, g:S^1\longrightarrow S^1$ given by $$f(z)=z^n\quad \textrm{and}\quad ...
0
votes
1answer
20 views

Show that if $h$ is extendable to a continuous map of $\Bbb R^n$ into $Y$, then $h_*$ is the trivial homomorphism.

Let $A$ be a subspace of $\Bbb R^n$; let $h:(A,a_0) \to (Y,y_0)$. Show that if $h$ is extendable to a continuous map of $\Bbb R^n$ into $Y$, then $h_*$ is the trivial homomorphism. I can't get any ...
3
votes
0answers
38 views
+200

Lifting a sphere-valued homotopy.

Let $A\subseteq X$ be two finite cell complexes, $\dim X\leq 2n-3$ and let $[(X,A), (S^n, *)]$ be the relative cohomotopy group. There is a natural map $$ \delta: [(X,A),(B^n,S^{n-1})] \to [(X,A), ...
1
vote
0answers
20 views

Geometrical Explanation of Borsuk Theorem

Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define ...
1
vote
0answers
36 views

fundamental group and covering space

I have a question about the fondamental group of the following covering space $$ p : Y \rightarrow X ; \; Y \owns (x,y) \mapsto x \in X $$ where $X = {\mathbb P}^1$ and $$ Y := \{(x,y) \in {\mathbb ...
3
votes
0answers
43 views

Short proof of Borsuk-Ulam's

By examining the singular cohomology ring with $\mathbb{Z}/2\mathbb{Z}$ coefficients, it is easy to see that if $n>m$ that there can be no map $f:\mathbb{R}P^{n}\to \mathbb{R}P^m$ that induces ...
0
votes
1answer
39 views

Fundamental group with unit circle and one axis removed

I am struggling with the following problem: Find the fundamental group of a space obtained by removing the unit circle from the x-y plane and z-axis from $\mathbb{R}^3$. I want to know what is the ...
0
votes
2answers
37 views

How to orient a manifold in the Euclidean space?

I learned that the orientation of a smooth manifold is a smooth choice of an orientation for bases of tangent space. Also I sometimes read that an embedded manifold in $\mathbb{R^3}$ inherites an ...
1
vote
1answer
20 views

Free loop space of classifying space as a disjoint union of classifying spaces of centralizer proof reference request.

I am looking for a reference for the proof or explanation of why for a discrete group $G$ we have that the free loop space of its classifying space is the disjoint union of centralizeers of $g$ where ...
3
votes
1answer
73 views

Another question in Bott-Tu.

I have a question regarding something on the Bott-Tu's book "Differential Forms in Algebraic Topology". At page 109, near the end, there is the following example: I don't manage to understand why ...
0
votes
1answer
24 views

A little question about contracting chain homotopy.

Let X be a topological space and $C=(C_n(X))$ be the singular complex. If there is a contracting chain homotopy, i.e. chain homotopy between $\text{id}_C$ and $0$, then $H_n(X)=0$. But I know that ...
1
vote
1answer
56 views

Covering map, singular homology

Let $X,Y$ be topological spaces and $q:Y\rightarrow X$ a covering map with $|q^{⁻1}({x} )|=n$ for all $x \in X$. I want to show that the induced map $$H_k(q,\mathbb{Q}):H_k(Y,\mathbb{Q})\rightarrow ...
-1
votes
0answers
21 views

A question about computing relative homology groups

I'm studying homology groups and there is something that I can't understand On the picture, the homomorphism from H1(A) to H1(X) is to be surjective, but I can't see why it is so. Could anyone ...
5
votes
1answer
75 views

Quantization of Chern number $c_1^n$ on 2n dimensional spin manifold

All orientable 2-manifolds are spin manifolds, and we know that the quantization of the first Chern number $c_1$ of a complex line bundle on 2-manifold is $\mathbb{Z}$. For 4-manifolds, the second ...
0
votes
0answers
37 views

Slice at a point of a topological space

The definition is from the following link -Slice at a point of a topological space Let $G$ be a topological transformation group of a Hausdorff space $X$. A subspace $S$ of is called a slice at a ...
1
vote
0answers
21 views

Barratt Whitehead Lemma, Proving exactness at the 'direct sum' module

I have been working out a proof of the Barratt Whitehead Lemma. Here it is. I had it all finished, but then I realised that I had assumed all the 'vertical' homomorphisms connecting the upper and ...
1
vote
1answer
27 views

Torsion in the Real Projective Plane

I'm trying to understand precisely why and how some closed curves are not boundaries while their "doubles" are. I know that taking any curve connecting antipodal points on $S^2$ projects to a closed ...
3
votes
0answers
52 views

Describing generators for the fundamental group of an elliptic curve given by an equation

Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a ...
0
votes
1answer
56 views

The Path Lifting Theorm for Local Homeomorphisms

We know that the path lifting theorem is true for covering maps. The map $p$ : $R_+$---->$S^1$ given by $p(t)$=(cos$2pi$t,sin$2pit)$ gives an example of a local homeomorphism which is not a covering ...
0
votes
2answers
61 views

Homotopic maps from $\mathbb{S}^{n} \to \mathbb{S}^{n}$

I'm trying to prove that if two (continuous) maps $f, g : \mathbb{S}^{n} \to \mathbb{S}^{n}$ are such that $f(x) \neq -g(x)$ for any $x \in \mathbb{S}^{n}$, then $f$ and $g$ are homotopic. But I ...
0
votes
0answers
20 views

An integrality theorem for immersions of complex projective spaces in the euclidean space

There are two questions: Please let me know your proof of the following theorem: If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to ...
1
vote
1answer
28 views

Adaptions I have to make to go from integer coefficients to coefficients in $R$

Let $R$ be a unital ring. I can prove that $\partial \circ \partial$ for the boundary map between singular homology groups with integer coefficients. Now I want to generalise to coefficients in $R$. ...
1
vote
1answer
57 views

On chain homotopy equivalence

I just learnt the notion of chain map and have the following question. Let $C=(C_n,\partial_n^C)$ and $D=(D_n,\partial_n^D)$ be chain complexes of abelian groups with boundary maps $\partial_n^C$ and ...
1
vote
0answers
28 views

Existence of covering space

I would like to know that if $X$ is a connected topological space, there is always a covering space of it, i.e., a continuous map $p:X'\to X$ with the known property.
1
vote
0answers
67 views

Characteristic class integral: on what manifold does $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$ hold?

Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds? Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd ...
3
votes
0answers
31 views

Presentation of group equal to trivial group

Problem: Show that the group given by the presentation $<x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2}>$ is equivalent to the trivial group. I have tried all sorts ...