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
23 views

How do i prove that this is homeomorphic to Klein Bottle?

My professor is teaching quotient space recently and he gives very informal arguments (drawing diagrams,arrows and stuff). I think he wants students to get familiar with quotient spaces and visualize ...
0
votes
2answers
31 views

A few questions regarding the winding number.

A few questions baout the winding numbers: Why do two homotopic paths have the same winding numbers? I think I can prove that two homotopic paths may have different winding numbers. Let $C$ be a ...
2
votes
0answers
38 views

Difference between two concepts of homotopy for simplicial maps?

I learn from Gelfand and Manin's Methods of Homological Algebra, Exercise 2 for I.4 that two maps $f,g\colon X\to Y$ between simplicial sets $X,Y$ are simply homotopic (maybe usually called simplicial ...
0
votes
1answer
21 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
1
vote
0answers
23 views

A Couple Formulas in Besse's “Einstein Manifolds”

In Besse's "Einstein Manifolds," Chapter 6D, there are 2 formulas which I am interested in, which apply to compact Riemannian $4$-manifolds: ...
0
votes
1answer
26 views

The signature of a product of surfaces

If $\Sigma_1$ and $\Sigma_2$ are surfaces (i.e. compact, oriented 2-manifolds without boundary), is the signature $\tau (\Sigma_1 \times \Sigma_2)$ well-known? Recall that the signature is the ...
1
vote
1answer
31 views

Fundamental crossed square of a square of spaces

I am a newbie at Topology and I didn't took the homotopy theory course. Sorry for that, but I need to know some facts about this paper ("Van Kampen Theorems for Diagrams of Spaces", authors R. Brown ...
2
votes
1answer
43 views

A condition for a covering map to be regular

We said that a path-connected covering map $p:E \rightarrow X$ is regular if: $\forall e \in p^{-1}(x_0): p_{\sharp} \pi_1(E,e)$ is a norm subgroup of $\pi_1(X,x_0)$. or equivalently: If closed ...
0
votes
1answer
31 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
5
votes
1answer
41 views

Orbit space of $S^n \times S^n$ under the antipodal action

Write $S^n$ for the $n$-dimensional sphere, the space of vectors of length $1$ in $(n+1)$-dimensional Euclidean space. Consider the antipodal action on $S^n$, i.e. the action of $\mathbb{Z}_2$ given ...
4
votes
1answer
67 views

Fundamental group of the complement of an eight shape

What is $\pi_1(\mathbb{R}^3 \setminus (S^1 \vee S^1))$? I would say that the space deformation retracts to a sphere $S^2$ surrounding the missing eight with two sticks stuck in the two loops of the ...
0
votes
1answer
21 views

Why are degree maps for cellular boundary formula from $S^{n-1}\to S^{n-1}$?

For a CW-complex, there's the cellular boundary formula that $$ d_n(e^n_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_\beta $$ where the coefficients $d_{\alpha\beta}$ are the degrees of the map $$ ...
0
votes
1answer
15 views

If $D^1\cup_f D^1=S^1$?

Suppose $f\colon S^0\to S^0$, so we can form the attaching space $D^1\cup_f D^1$. Is my intuition correct that this space is just $S^1$? Since $S^0=\{1,-1\}$, $f$ is either the identity, or swaps the ...
3
votes
0answers
52 views

About homotopy fiber at Hatcher's book

What is the meaning of the statement: In this case a map $(I^{i+1},∂I^{i+1},J^i) \to (B,A,x_0)$ is the same as a map $(I^i,∂I^i) \to (F_f, \gamma_0)$ where $\gamma_0$ is the constant path at ...
0
votes
2answers
60 views

About the definition of homology

can someone explaine me this definition of Homology: "The homology groups of $X$ measure "how-far" the chain complex associated to $X$ is from being exact." I know that homology measure the number ...
0
votes
1answer
46 views

Is the pullback of two covering spaces $\tilde X$ and $\hat X$ a covering space?

Suppose we have two covering spaces $p:\tilde X \rightarrow X$ and $q:\hat X \rightarrow X$ of the same space. Is the pullback $\tilde X \times_X \hat X$ also a covering space of $X$? If yes, what ...
3
votes
1answer
84 views

Questions about simplex and affine space

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
5
votes
0answers
177 views

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in ...
0
votes
2answers
34 views

How do we prove that the fundamental group is a group?

My understanding of the fundamental group is that it's the set of all loops starting and ending at a point $x_0$ in a space $X$, along with the operation of composition. For it to be a group, ...
0
votes
0answers
50 views

Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
3
votes
1answer
27 views

Constructing a surjection from fundamental group of a mapping cone to Hawaiian Earring to $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$

If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X = $ (the Hawaiian Earring) which ...
0
votes
2answers
21 views

retraction induced homomorphism is surjective

Im having a hard time proving this although it looks trivial... Let $r:X\to A$ be a retraction between a topological space $X$ and $A\subset X$ such that $r(a_0)=a_0$ for $a_0\in A$ then the induced ...
2
votes
0answers
40 views

Homomorphisms induced by maps $S^1 \times S^1 \rightarrow S^1 \times S^1$.

Problem 2.2.30 in Hatcher involves the homomorphisms $H_2(S^1\times S^1) \cong \mathbb{Z} \rightarrow H_2(S^1\times S^1) \cong \mathbb{Z}$ induced by The map $S^1 \times S^1 \rightarrow S^1 \times ...
2
votes
1answer
49 views

A question on finite non-contractible CW complexes

The algebraic topology book I am reading recently covered the following theorem named after Whitehead and corresponding direct consequence. THEOREM. If X is a CW complex of dimension less than n and ...
1
vote
1answer
67 views

Is the suspension of a countable collection of points in $\mathbb{R}$ a countable collection of circles?

I am extremely new to topology and taking an algebraic topology course, and I need some help understanding the behavior of suspensions. The problem I am working on asks about the suspension of the ...
2
votes
0answers
42 views

Homotopic paths implies equal winding numbers

I am trying to prove a proposition relating analysis and geometry. I have a general idea on how to prove it. However, a small part of the proof needs a lemma about path homotopy and winding number. ...
3
votes
2answers
74 views

Any relations between the weak topology on a Banach Space and the weak topology on CW complexes?

I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, ...
1
vote
1answer
33 views

Maps from cogroups to groups & Eckmann-Hilton

One way to prove that a topological group has abelian fundamental group is to point out that the two group operations are homomorphisms for each other and apply the Eckmann-Hilton argument. ...
7
votes
2answers
60 views

When does a continuous map $f:X\rightarrow \mathbb{H}P^n$ lift to $S^{4n+3}$?

Consider the Hopf bundles $$S^1\rightarrow S^{2n+1}\rightarrow \mathbb{C}P^n$$ and $$S^3\rightarrow S^{4n+3}\rightarrow \mathbb{H}P^n.$$ In this question (and also here), it is shown that for any ...
1
vote
2answers
38 views

surface presentation

Given the following group and presentation, how could I go about showing if there exists a compact surface with that fundamental group? The group is $\big \langle a, b, c, d, e $ $\mid$ ...
2
votes
1answer
29 views

Is there a name for spaces that always have local sections?

Given a continuous map $p:E \rightarrow B$ Suppose for every point $b \in B$ and a point $x \in p^{-1}b$ in the fibre of it, there is an open set $V$ of $B$ that contains the point $b$ such that ...
2
votes
0answers
54 views

cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
4
votes
1answer
28 views

Linearly Independent Curves on Genus g Surface

I have seen the following claim: Let $\gamma_1, \gamma_2, \cdots, \gamma_g$ be a collection of $g$ non-intersecting, simple closed curves on a genus $g$ surface, $\Sigma$. Then the $\gamma_i$ are ...
2
votes
0answers
29 views

Module structure in the Serre spectral sequence of the Borel construction

Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its ...
2
votes
1answer
43 views

Self-intersections of loops

Let $X$ be a topological space with basepoint $x_0$. Define a map $s\colon\pi_1(X,x_0)\to\mathbb{Z}_{\geq 0}$ by $$s([\gamma])=\min\{\text{number of self-intersections of }\gamma'\colon \gamma'\in ...
8
votes
1answer
132 views

Modern Research in Algebraic Topology

What are some of the main directions and trends in modern (let's say within the last ~10 years) algebraic topology? What are some major open problems or recent results? In a more specific direction, ...
4
votes
2answers
111 views

Lifting cohomology-killing maps through the 3-sphere

In his first answer to this question, Jason deVito claimed that a map $f:X\to S^2$ kills $H^2$ if and only if it factors through the Hopf fibration $\pi:S^3\to S^2$. What's the justification for this ...
1
vote
0answers
20 views

Left and right module on the cohomology of a sheaf

Let $X$ a topological space, say a complex variety, and $\mathbb{C}_X$ its constant sheaf. $\mathcal{D}(X)$ is the derived category of sheaves of $\mathbb{C}_X$-modules. Let $F^\bullet\in ...
1
vote
1answer
62 views

Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
0
votes
1answer
28 views

Identification of points versus line drawn between points

I have a question regarding fundamental groups. If I take a sphere and union a line between it's poles, is that the same space as the sphere with those poles identified? I am trying to find the ...
0
votes
1answer
16 views

homeomorphism classes of compact surfaces with addition operation is a monoid

This is essentially pg 6 of serge lang's algebra's discussion about an interesting example. Homeomorphism classes of compact surfaces with the addition operation defined as following. Say M and $M'$ ...
2
votes
3answers
118 views

More elementary proof that $\pi_n(S^n) \cong \mathbb{Z}$

The proof I know that $\pi_n(S^n) \cong \mathbb{Z}$ is based on the Hurewicz theorem (which implies that $\pi_n(S^n) \cong H_n(S^n)$). I'm looking for a more elementary argument - preferably ...
1
vote
1answer
30 views

Étalé space for sheaf of sections of a fiber bundle

Let $X$ be a topological space, $\pi:E\to X$ a fiber bundle over $X$ with fiber $F$ and structure group $G$. Let $\mathcal{F}$ denote the sheaf of continuous sections of the bundle. I probably want to ...
0
votes
1answer
19 views

Complex bundles on $S^{2n+1}$

We know that the complex $K$ theory of spheres are 2 periodic. On the other hand every complex bundle on $S^{1}$ is trivial. So $K(S^{2n+1})=0$. So this is a motivation to ask: Is there a ...
1
vote
1answer
47 views

Real vector bundles on $S^{7}$

Is it true that $\pi_{6}(O(n))=0$ for all n? Equivalently, are all real bundles on $S^{7}$ trivial?
4
votes
1answer
46 views

map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
0
votes
1answer
28 views

How to find a sequence in discrete group

Let $\Gamma$ be a discrete group. Can we find an increasing sequence $F_{n}\subset \Gamma$ of finite subsets, such that $\cup F_{n}=\Gamma$?
0
votes
1answer
33 views

A question about winding numbers.

This is a question from Needham's "Visual Complex Analysis". Kindly refer to the photo below. Let $K$ be a line moving downwards. The book says that if we move a point $r$ from the left to the ...
13
votes
4answers
173 views

Are there nontrivial continuous maps between complex projective spaces?

Are there maps $f: \Bbb{CP}^n \rightarrow \Bbb{CP}^m$, with $n>m$, that are not null-homotopic? In particular, is there some non-null-homotopic map $\Bbb{CP}^n \rightarrow S^2$ for $n>1$? Can we ...
4
votes
1answer
35 views

Converse to the Jordan-Brouwer separation theorem

By the Jordan curve theorem, if $C \subset S^2$ is (the image of) a simple closed curve, then $S^2 \setminus C$ has precisely two connected components. This statement admits the following "converse". ...