0
votes
0answers
22 views

invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of "Homotopy invariance of line integral ...
1
vote
0answers
35 views

Is nerve theorem always true?

Is the nerve theorem true for not paracompact spaces? Background: Nerve theorem states that if $U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many ...
1
vote
1answer
48 views

Every Group is a Fundamental Group

I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and ...
0
votes
1answer
35 views

What is the map $\Sigma K(G,n) \to K(G,n+1)$?

Since $\Omega K(G, n+1)$ is a $K(G,n)$, we have a CW approximation/homotopy equivalence $K(G,n) \xrightarrow{\sim} \Omega K(G,n+1)$. The adjoint of this map is a map $\Sigma K(G,n) \to K(G,n+1)$. ...
3
votes
0answers
47 views

Definition of the triad homotopy groups

Let $ \ n \in \mathbb{N}$, $n \geqslant 2$. Equip $\mathbb{R}^n$ with the usual euclidean norm $ \ \Vert . \Vert : \mathbb{R}^n \to \mathbb{R}$, metric and topology. Consider $ \ p_0 = (1,0,0,...,0,0) ...
0
votes
0answers
24 views

n-connected pair (X,A) implies that the inclusion A to X is an n equivalence

I apologize if this is a novice question but I wanted to get some details on this concept. The algebraic topology book I am reading says that a relative CW complex $(X,A)$ is n-connected when ...
1
vote
0answers
18 views

Injectivity of a map from a homotopy set to a homology group

Let $\Sigma$ be a closed Riemann surface and $X$ a 1-connected topological space. I would like to prove the following fact. (A) The map $[\Sigma,X] \to H_2(X;\mathbb Z)$ defined by $[f] \mapsto ...
3
votes
1answer
45 views

Understanding Hatcher's proof of Borsuk-Ulam theorem for $n=2$

I am trying to understand the proof of the Borsuk-Ulam theorem for $S^2$ given in Hatcher's "Algebraic Topology" (Th. 1.10), as another person does here, but we are stuck at different places, so I ...
0
votes
1answer
36 views

homotopy type not constant during a homotopy

What is a possibly easy example of a topological space $X$ and a homotopy $H:X\times I\to X$, $H(x,0)=x$ for all $x\in X$, such that the homotopy type of the subspace $h_t(X)=H(X,t)$ is not constant ...
2
votes
0answers
39 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 ...
1
vote
1answer
33 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 ...
3
votes
0answers
53 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 ...
2
votes
1answer
50 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 ...
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
1answer
63 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 ...
13
votes
4answers
175 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 ...
5
votes
1answer
56 views

Homotopical classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$

Which are homotopy classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$ for $n < m$? In real case, even for any cellular complex $X$ with $\dim X<m$ homotopy classes of mappings $X \to ...
3
votes
0answers
30 views

Can one prove that $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$ without invoking the long exact sequence of a fibration?

In trying to remind myself why $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$, the natural argument that comes to mind is to take the long exact sequence associated to the fibration $S^1 \rightarrow S^\infty ...
0
votes
1answer
27 views

smash product of Eilenberg-Maclane spaces

Let $G$ be an abelian group and $K_n=K(G,n)$ be the Eilenberg-Maclane space. How to obtain $K_m\wedge K_n$ is $(m+n-1)$-connected? (Hatcher's book page 404)
2
votes
1answer
45 views

Is the homology theory given by Eilenberg Maclane spectrum equal to ordinary homology?

(I think I'm missing something very simple). Let $R$ be a ring and $HR$ the associated Eilenberg-Maclane spectrum, defined by $$[\Sigma^\infty_+ X,HR]_{-*}={H}^*(X; R)$$ for any CW-complex $X$, and ...
2
votes
3answers
52 views

homotopy groups of wedge sum

Let $X_\alpha$ be connected CW-complexes. Then from Hatcher's book, $$\pi_{n}(\prod_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha}).$$ Is it true in general $$\pi_{n}(\bigvee_{\alpha} ...
1
vote
2answers
47 views

$X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point

Im trying to show that: for $X,Y$ topological spaces $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point while $[X,Y]$ denote the set of homotopy classes of maps of $X$ ...
2
votes
3answers
58 views

Hatcher, p.35, $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$

Could some help me understand how $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$. Also, if the one point set is replaced by any finite set, how does the argument work. This ...
2
votes
1answer
54 views

Motivation for the proof of the associativity of multiplication of equivalence classes of paths

After having defined the equivalence classes of paths in a topological space in chapter two of the book A Basic Course in Algebraic Topology, William S. Massey proves the lemma The multiplication ...
2
votes
2answers
66 views

prove that the identity map $i:S^n\to S^n$ is not nullhomotopic.

Using the fact that: For all $n\in \mathbb{N}$ there is no retraction $r:B^{n+1} \to S^n$, prove that the identity map $i:S^n\to S^n$ is not nullhomotopic. This is a problem in section 56 of Munkres' ...
2
votes
4answers
26 views

$X$ is contained in an annulus containing a circle in the annulus.Show that $\pi_1(X)$ contains a subgroup isomorphic to $\mathbb{Z}$.

In $\mathbb{R^2}$ let $C=\{|x|=2\}$ and $A=\{1<|x|<3\}$, let $X$ be a path connected set such that $C \subset X \subset A$. Show that $\pi_1(X)$ contains a subgroup isomorphic to $\mathbb{Z}$. ...
2
votes
2answers
68 views

Homotopy direct limit versus direct limit

Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and ...
3
votes
1answer
37 views

Does the $\Omega$-spectrum functor send exact triangles to homotopy cofiber sequences?

The functor $\Omega^\infty\colon Spectra\to Spaces$ which takes a spectrum, replaces it by the associated $\Omega$-spectrum and then takes its $0$th space sends exact triangles to homotopy fiber ...
0
votes
0answers
32 views

A curve not homotopic to constant path but index of every point is zero.

I want to find a curve which is not homotopic to constant path but the index of every point not on the curve is zero. Here the domain is an open subset of Complex Plane.I was unable to find any such ...
2
votes
1answer
34 views

Extending a homeomorphism of the open disk to the boundary.

Let $D^2 = \{x \in \mathbb{R}^2 : ||x||\leq 1\}$ denote the closed disk and $int(D^2)$ denote its interior. If I have a homeomorphism $\ f: int(D^2) \rightarrow int(D^2)$ it is clear that it is not ...
1
vote
0answers
33 views

why does the flipping map $S^1 \wedge S^1 \to S^1 \wedge S^1$ introduce a minus sign in homotopy?

Suppose we have a map $f:S^1 \wedge S^1 \to X$ of pointed spaces ($S^1$ is the circle), let $T:S^1 \wedge S^1$ be the map that flips the factors, (so $T(x,y)=(y,x)$) and let $f'=f \circ T:S^1 \wedge ...
0
votes
1answer
49 views

Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?

Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$? For reasonable spaces $X$, e.g. CW-complexes, one has ...
1
vote
2answers
58 views

Proof of the Borsuk-Ulam Theorem

The Borsuk-Ulam Theorem says the following: For any continuous map $g: S^n \rightarrow \mathbb{R}^n$ there exists $x \in S^n$ such that $g(x)=g(-x)$. I'm trying to work through the proof given in ...
0
votes
1answer
47 views

Is the homotopy class given by the degree?

Let $X$ be a topological space such that $\pi_n(X)=H_n(X)=Z$. A continuous map $f: S^n \rightarrow X$ is an element of $\pi_N(X)=Z$ therefore $[f]_{\mathrm{homotopy}}$ is characterised by an integer ...
1
vote
0answers
36 views

Loop space of $S^1$

How concretely can the (based) loop space $\Omega S^1$ of $S^1$ be described? I know it's a space with homotopy groups $\pi_0(\Omega S^1) \simeq \mathbb{Z}$ and $\pi_i(\Omega S^1) \simeq 0$ for ...
4
votes
0answers
137 views

Higher homotopy groups: Basepoint independence.

Let $f,g: [0,1]^n=I^n \rightarrow X$ be contiuous maps s.t. $f(\partial I^n)=g(\partial I^n)=x_1$. If $\gamma:I \rightarrow X$ is a path joining $x_0$ and $x_1$ ...
1
vote
1answer
40 views

cohomology is dual to homology of a spectrum if homology is free

Let $E$ be a multiplicative spectrum (and $X$ a space with $H_n(X; \mathbb{Z})$ free abelian for every $n$). The following excerpt is taken from the notes here claim that item (1) below easily implies ...
0
votes
1answer
48 views

Exercise 2, chapter 4, Hatcher.

Show that if $\varphi: X \rightarrow Y$ is a homotopy equivalence, then the induced homomorphisms $\varphi_{*}:\pi_n(X,x_0) \rightarrow\pi_n(Y,\varphi(x_0))$ are isomorphisms, for all n$\in ...
0
votes
0answers
57 views

Intuition behind certain examples of fundamental groups

I have some intuition behind the interpretation of having nontrivial fundamental group, detecting the holes in the space and so on. But I don't quite see how interpret the fact that the fundamental ...
3
votes
0answers
42 views

definitions of various spectra: $E^X$ and $E \wedge \Sigma^\infty X$

Let $E=\{E_n\}$ be a spectrum given by a sequence of pointed CW complexes $E_n$ and inclusions $\Sigma E_n \to E_{n+1}$. Let $X$ a pointed CW complex. I had a few very naive questions I had while ...
1
vote
2answers
153 views

A doubt in Hatcher's Algebraic Topology.

I refer to pg. 27 of Hatcher's Algebraic Topology. I refer to the part where Hatcher proves that $f.(g.h)\cong (f.g).h$ For the life of me, I cannot figure out how the diagram on the right proves ...
4
votes
1answer
45 views

Homotopy classes of maps from the projective plane to $S^1 \times S^3$

I have a past qual question here: characterize the space $[(\mathbb{RP}^2,x),(S^1 \times S^3,y)]$ of homotopy classes of maps from $(\mathbb{RP}^2,x)$ to $(S^1 \times S^3,y)$, where here $x \in ...
3
votes
1answer
49 views

Homotopy groups relating to toric varieties

It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, ...
2
votes
1answer
33 views

The cone minus its apex deformation retracts onto its basis

Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$. My ...
2
votes
0answers
70 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
6
votes
2answers
117 views

Is this Space Homotopy Equivalent to $S^2$

Let $X$ be the space $S^1$ with two $2$-cells attached via maps of relatively prime degrees. This space is simply connected and has the homology of $S^2$, but is it homotopy equivalent to $S^2$?
1
vote
0answers
41 views

Homotopy groups of unitary groups

in this paper I found some explicit generators of homotopy groups of unitary groups, for example $\pi_3[SU_2]$: $\begin{bmatrix}z_1\\z_2\end{bmatrix}$$\rightarrow $$\begin{bmatrix}z_1 ...
1
vote
0answers
35 views

$\pi_0$ of $M(2) \wedge M(2)$

My motivation is trying to understand Tom Goodwillie's argument here: http://mathoverflow.net/questions/87919/difficulties-with-the-mod-2-moore-spectrum and the only thing I don't get is why ...
2
votes
2answers
36 views

Expressing homotopy groups of spaces of (unpointed) maps $S^1\to M$ in terms of homotopy groups of spaces of pointed maps.

I came across the following problem while studying for a topology exam: Let $M$ be a topological space, let $\Lambda(M)=M^{S^1}$, the space of continuous maps $S^1\to M$ with the compact-open ...
6
votes
1answer
182 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...