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

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2
votes
1answer
50 views

Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
1
vote
0answers
21 views

Categorical Notion of Quotient in Spectra

Having done some reading on spectra recently, I noticed that the definition of a quotient spectra for a closed subspectrum of a CW spectrum is simply given by taking the quotient of each of the spaces ...
8
votes
2answers
183 views

Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?

I am curious if there is a decent "bare hands" proof that the fundamental group of $S^1$ is $\mathbb Z$ that does not invoke covering space theory. One must show two claims. First, that $f(t)=e^{2\pi ...
2
votes
0answers
43 views

Different groups with the same classifying space.

Let $G$ be a topological group and $BG$ its classifying space. From the LES of the universal bundle, we get $\pi_i(BG)\cong\pi_{i-1}(G)$, so given the classifying space, we know all homotopy groups of ...
2
votes
1answer
36 views

A compact Lie group modulo by its maximal torus has nonzero Euler characteristic

In Andrew Baker's Matrix Groups, (in the proof of Theorem 20.11), there is an unproven statement that if $G$ is a compact Lie group and $T$ is a maximal torus, then $\chi (G/T)\ne 0$. I have an ...
5
votes
0answers
34 views

Homeomorphism between boundaries of open sets is “surjective”

Let $U,V\subset\mathbb{R}^n$ two bounded open sets and let $F:\overline{U}\to\mathbb{R}^n$ be a continuous map. Assume that $F$ maps $\partial U$ homeomorphically onto $\partial V$. The question: is ...
0
votes
2answers
110 views

Inclusion induces identity on homology

Let $(H_*, \partial_*)$ be a homology theory with values in the category of $\Bbb{Z}$-modules satisfying the dimension axiom. Then the inclusion $S^1\vee S^1\to T^2$ should induce (up to isomorphism) ...
0
votes
0answers
14 views

Local homology of a fibred product

Let $A,B$ be topological spaces and suppose that for $a\in A$ and $b\in B$ the local singular homology groups $H_k(A,A\setminus\{a\};\mathbb{Q})$, $H_k(B,B\setminus\{b\};\mathbb{Q})$ are known for all ...
0
votes
0answers
19 views

$ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $is a principal $ O(n-k) $-bundle.

I'm trying to prove that $ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $; $ A \longmapsto (Ae_1, ... ,Ae_k) $ (the projection from the orthogonal group to the Stiefel manifold) is a ...
1
vote
0answers
41 views

finite dimensional CW complex

Let $X$ be a finite dimensional CW complex, where X is simply connected and at least one $H_{i}(X)$, is non trivial (so that X not be contractible). Can we conclude that X has at least one non trivial ...
1
vote
0answers
45 views

Application of Morse Inequalities

I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ...
12
votes
2answers
125 views

Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

$\DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb{R}}$ If $f,g : X \to Y$ are two maps (all maps considered are continuous here), a homotopy between $f$ and $g$ is a map $H : [0,1] \times X \to ...
1
vote
1answer
35 views

what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
1
vote
1answer
124 views

Fundamental group of $ \mathbb{S}^{n-1}\times\mathbb{R}$ minus $k$ disks $\mathbb{D}^n$

Let $X$ be the space obtained from $ \mathbb{S}^{n-1}\times\mathbb{R}$ by deleting $k$ disjoint subsets, each one homeomorphic to $D^n$. What is the foundamental group of $X$?
2
votes
1answer
38 views

Graph embedding into a surface

For example, let's consider a $K_{5}$ (complete graph on 5 vertixes) and a torus, which is defined as $S^{1} \times S^{1}$. How to build a continous embedding $f:K_{5} \rightarrow \mathbb{T}^{2}$? We ...
4
votes
2answers
61 views

If $G$ is a finite nontrivial group, then $K(G, 1)$ cannot be a finite CW-complex?

I am currently working on this algebraic topology problem and got stuck: Suppose $X$ is a finite CW complex with $\pi_1(X)$ a nontrivial finite group. Show that its universal cover $\widetilde{X}$ ...
8
votes
1answer
98 views

Does every continuous action of $S^1$ on $R^n$ have a fixed point?

I certainly can't think of one that doesn't. I am aware that there are decompositions of $R^n$ as a union of embedded $S^1$'s, but none of these seem like they would support a continuous action. ...
2
votes
1answer
41 views

Find a presentation for the fundamental group of $P^2\#T$

I have to find a presentation for the fundamental group of $ P^2\# T $. Which I believe to be identified by the following labeling scheme: $(a_1b_1a_1b_1)(a_2b_2a_2^{-1}b_2^{-1})$. $P^2$ is the ...
0
votes
1answer
18 views

Weak equivalence of ordinary and homotopy colimits

I am looking for conditions under which colimits and homotopy colimits of diagrams of, say, topological spaces, are weakly equivalent. I would appreciate answers not demanding an all too profound ...
0
votes
1answer
133 views

canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization ...
5
votes
1answer
78 views

Ideas for basic application of homotopy theory to homological algebra?

I'm taking a first course in homological algebra. As a project, the lecturer suggested each student find a topic, presentable in an hour, relating to the material studied in the course. The material ...
5
votes
1answer
86 views

Is there a compact manifold having Euler characteristic 0 which cannot be given a Lie group structure?

I realized that a (compact) Lie group must have Euler characteristic 0 due to Poincare-Hopf index theorem. Now I'm thinking of its converse. Is there a compact manifold having Euler characteristic 0 ...
2
votes
1answer
43 views

underlying real vector bundle of a complex vector bundle

Let $\eta^\mathbb{C}$ be a complex line bundle. If the underlying $2$-dimensional vector bundle $\eta$ is not trivial as a real vector bundle, can we obtain that $\eta^\mathbb{C}$ is not trivial as a ...
4
votes
1answer
115 views

Homology of a co-h-space manifold

Let $M$ be a compact connected topological manifold of dimension $n>1$. Suppose the corepresented functor $[M,-]\colon Top_{\ast}\rightarrow Set$ lifts to monoids or equivalently that $M$ is a ...
3
votes
0answers
31 views

Proof of Alexander Duality on Bredon - help with a passage

At page 353 of Bredon's Topology and Geometry, there is stated the Alexander Duality as Corollary $8.7$. I don't understand where does the upper row come from and why is it exact. I thought it was ...
6
votes
1answer
76 views

knot theory: two definitions of equivalence (ambient isotopy and homeomorphism)

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
2
votes
0answers
42 views

Homotopy colimits preserve weak equivalences

It is well known that homotopy colimits of diagrams are constructed so that if one has weak equivalences between all objects of two diagrams (under the same indexing category) the induced map between ...
1
vote
1answer
74 views

what is the classifying space of a monoid

In the paper Homology Fibrations and the "Group-Completion". Theorem. McDuff, D.; Segal, G., 1976, the first line: A topological monoid $M$ has a classifying space $BM$. I do not understand this ...
0
votes
2answers
36 views

Legitimate homotopy between $(\gamma_1 * \gamma_2) *\gamma_3$ and $\gamma_1*(\gamma_2 * \gamma_3)$?

I want to find a homotopy between curves $(\gamma_1 * \gamma_2) *\gamma_3$ and $\gamma_1*(\gamma_2 * \gamma_3)$ that are closed loops in some topological space $X.$ I found $H(t,s):=$ $ \gamma_1 ...
1
vote
1answer
41 views

If $F$ a sheaf and $S\subset F$ a subfunctor, then $S$ is a subsheaf if and only if…

This is Proposition 1 from Maclane & Moerdijk's Sheaves in Geometry and Logic, part II, section 1. Proposition 1. Let $F$ be a sheaf on $X$ and $S\subset F$ a subfunctor. $S$ is a subsheaf if ...
2
votes
2answers
34 views

Triangulation of the projective plane

I just worked a little bit with triangulations of surfaces. I think the following "triangulation" of the real projective plane is false: The red (blue) edges are identified in an inverse way. Sorry ...
1
vote
0answers
35 views

Does pushing point along a loop on a surface induce a homotopy from identity to a homeomorphism of the surface?

Suppose $S$ is a closed surface (assume $\chi(S)\leq 0$ if necessary). Let $p$ be a fixed point on the surface $S$, and $\beta$ be an oriented loop based at $p$. Intuitively, I can put finger on $p$ ...
4
votes
0answers
40 views

Definitions of the ring $R/(x_0^\infty,\ldots,x_{n-1}^\infty)$

Let $R$ be a ring, and let $I = (x_0,\ldots,x_{n-1})$ be a finitely-generated ideal inside of $R$, generated by a regular sequence. In algebraic topology one often encounters a ring, usually denoted ...
3
votes
1answer
50 views

Direct Limit in HTop

I am currently trying to figure out the importance of homotopy direct limits in Top, which are of course different from direct limits in HTop. I have been told that the latter need not even exist, but ...
0
votes
0answers
51 views

prove that there is not a homeomorphic between $E$ and $\mathbb{R}$

Prove that $X=\{(x,y) \in \mathbb{R^2}: ((x-2)^2+y^2-1)\cdot(x^2+y^2-1)=0 \}$ has a universal cover $(E,p)$ and prove that $E$ is not homeomorphic to $\mathbb{R}$ $X$ is a bouquet of two circles and ...
4
votes
4answers
226 views

section of a fiber bundle

I heard in class that not every fiber bundle admits a section. I am not sure why this is true, you can always pick a point on a fiber and follow it through as you glue local trivializations then you ...
2
votes
2answers
62 views

A space $X$ with $S^{2n+1}$ as universal covering space must be orientable.

Problem If $X$ has $S^{2n+1}$ as universal covering space, then show that $X$ must be orientable. My idea: By contradiction, suppose $X$ is non-oreintable. Then we consider the orientation covering ...
1
vote
1answer
62 views

Computing homology w/ Mayer Vietoris

Let $(H_*, \partial_*)$ be a homology theory satisfying the dimension axiom. Let $n\ge1$ and $X= S^n \cup_f D^{n+1}$ where $f:S^n\to S^n$ is a degree $k$ map. Compute each homology group of $X$. I ...
0
votes
0answers
44 views

Is this a functor?Is this a useful functor?

To every topological space $X$ we associate a sequence of groups $G_{n}(X)$ as follows: There is a natural morphism $\alpha_{n}:\pi_{n}(X)\to \pi_{n+1}(SX)$ which maps each $\gamma:S^{n}\to X$ to ...
1
vote
1answer
47 views

If $(X,A)$ has homotopy extension, then $X \times I$ def. retracts to $X \times \{0\} \cup A \times I$

Exercise 0.26 in Hatcher's Algebraic Topology is Use Corollary 0.20 to show that if $(X,A)$ has the homotopy extension property, then $X \times I$ deformation retracts to $X \times \{0\} \cup A ...
2
votes
2answers
54 views

Find the fundamental group of $X$

Let $X$ be the unit square with corners identified. I was thinking about its fundamental group. My strategy was to visualize it as e CW complex with a single $0$-cell, four $1$-cells (i.e a wedge of ...
2
votes
0answers
56 views

Formula for Stiefel-Whitney of tensor product

I need help for solving Ex. 7C from 'Characteristic Classes' by Milnor/Stasheff: The exercise asks to find a formula for the (total) Stiefel-Whitney class of $\xi^m\otimes\eta^n$ over a paracompact ...
1
vote
0answers
45 views

$SO(3)$ vs 3-Torus

$SO(3)$ and 3-Torus both can be viewed via rotations for a rigid body. They are not diffeomorphic. $SO(3)$ can be decomposed into three axial rotations. Could I think the reason they are not ...
3
votes
2answers
61 views

Whether or not such a simple CW complex can be made a $C^{\infty}$ manifold?

Problem Let $X$ be the space obtained by attaching two disks to $S^1$, the first disc being attached by the 7 times around,i.e. $z \to z^7$, and the second by the 5 times around. Can $X$ be made ...
2
votes
1answer
35 views

Alexander polynomial of unknot without Fox calculus or infinite cyclic cover

As explained in Lickorish`s book "Introduction to knot theory", one can define the Conway-normalized version of the Alexander polynomial by the determinant of certain sum of Seifert matrix plus ...
2
votes
0answers
36 views

There is no quasiconformal map from punctured ball into ring (on the plane)

The idea is to use l2 cohomology as a quasiregular map invariant. It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form in punctured ...
5
votes
2answers
75 views

Relations on Stiefel-Whitney classes

Can arbitrary cohomology classes $w_1,\dots,w_n$ from $H^{*}(B,\mathbb{Z}_2)$ be Stiefel-Whitney classes of some bundle over the given base $B$ or there are some necessary relation which can be ...
2
votes
1answer
65 views

Mayer-Vietoris in reduced homology for a torus.

By using the Mayer-Vietoris sequence in reduced homology : I have to calculate the homology groups of : The torus $\mathbb{T}^2 :=[0;1]^2 /\mathcal{T}$ by using the following decomposition $X_1 := ...
3
votes
1answer
50 views

Calculate the cohomology group of $U(n)$ by spectral sequence.

Here $U(n)$ is the unitary group, consisting of all matrix $A \in M_n (\mathbb{C})$ such that $AA^*=I$ Problem How to calculate the integer cohomology group $H^*(U(n))$ of $U(n)$? What if $O(n)$ ...
-1
votes
0answers
24 views

How to prove a diameter formula for a specific simplicial partition?

Let $S=<<x^0,x^1,...,x^k>>$ be a $k$-dimensional simplex in $\mathbb R^n$, and let $\mathcal T$ be a simplicial partition of $S$. Let $y \in\mathbb R^n$ be a vector that is affine ...