Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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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 ...
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28 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
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41 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 ...
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52 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 ...
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37 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 ...
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49 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
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2answers
138 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 ...
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36 views

Can all null-homotopy be made differentiable on arbitrary metric space?

Let $M$ be a metric, and assume that it is simply connected. For a closed curve $f$, we define it to be differentiable iff for any $x$ then $\lim\limits_{h\rightarrow 0}\frac{d(f(x),f(x+h))}{h}$ ...
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43 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 ...
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43 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, ...
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93 views

Realization (in the sense of homotopy coherent nerve) of $\partial\Delta^n$

I need some help understanding the working of the homotopy coherent nerve (as described in Lurie's HTT). Let $i < j$, then $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) \simeq (\Delta^1)^{(j-i-1)}$, ...
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$C_g \simeq SX$ and $C_h \simeq SY$ [closed]

Hi need some help with this problem: Let $f : X \to Y$ . Then we can form the cofiber sequence $X \to Y \to C_f \to C_g \to C_h$ where $g: Y \to C_f$, $h: C_f \to C_g$, and $i: C_g \to C_h$. Show ...
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1answer
31 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 ...
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48 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 ...
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13 views

Homotopy for the complement when slightly thickening a subspace of $ℝ^n$

For a given subspace $X \subset ℝ^n$ there is a homotopy equivalence $X \times \{0\} \simeq X \times I$ (subspaces of $ℝ^{n+1}$), where $I$ is the unit interval. However, the complements in $ℝ^{n+1}$ ...
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110 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 $S^2$?
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40 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 ...
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34 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 ...
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35 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 ...
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51 views

Homotopy of Involutory Matrices?

I want to construct a homotopy from the matrix $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ ...
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1answer
141 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 ...
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27 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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23 views

Being smooth homotopic relation: proof

Suppose we have an open set $U$ in the plane and two $\cal C^\infty$ paths $\gamma,\eta:[a,b]\to U$ with the same endpoints (i.e., $P:=\gamma(a)=\eta(a)$ and $Q:=\gamma(b)=\eta(b)$). We say that ...
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101 views

How an empty set is collapsed to a point?

In the original book of Conley Index Theory: Isolated Invariant Sets and the Morse Index chp3.3, p6, Charles Conley mentioned that ...
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1answer
28 views

What is the meaning of “Homotopy of Little disc Operads”

I want to understand what means the homotopy of the little discs operad. I'm starting to research in this area and I have some questions. 1) I don't understand why little discs operad is a ...
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2answers
75 views

Topological/homotopical classification for 1-dim CW-complexes?

It's a common exercise to classify a collection of 1-dim objects, say the figures of 0-9, or A-Z, up to homeomorphism or homotopy equivalence. I suddenly raise a question in general: Is there any ...
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26 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
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27 views

Showing this Null homotopic composite factors through a Null homotopic map

I was having some trouble with this concept which makes sense to me intuitively but the understanding of which is not yet fully clear. Suppose $CS^n$ is the unreduced cone on the n-sphere $S^n$. By ...
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124 views

Do freely homotopic maps induce the same homomorphism on fundamental groups?

Let $f,g\colon X\to Y$ be two continuous maps that are freely homotopic, such that there is some $x_0\in X$ with $f(x_0)=g(x_0)$. Is it true that the induced homomorphisms $f_*,g_*\colon ...
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31 views

Proof the Associative Property of H-group

I would like to prove the following proposition: Let $X$ and $Y$ be based topological spaces and let $[X,Y]$ be the set of homotopy classes of based maps $X\to Y$. If for every $X$, $[X,Y]$ is a ...
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1answer
100 views

Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
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52 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
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73 views

Homotopy equivalent iff isomorphic homotopy groups?

Is it true that two spaces or $\infty$-groupoids are homotopy equivalent if and only if they have isomorphic homotopy groups?
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150 views

Can we simultaneously realize arbitrary homotopy groups and arbitrary homology groups?

Let's keep our groups finitely presented for the time being. All spaces in this post are path connected. Background: By a standard construction (e.g., on p. 365 of Hatcher), there exists a $K(\pi, ...
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65 views

Homological algebra (homotopical approach)

I have gone through a couple of courses in homological algebra, in the context of derived functors, abelian categories,... Now I would like to watch it from another perspective: my main interest is ...
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30 views

Showing that a homotopy fiber of a fibration is homotopy equivalent to the fiber of the base point.

Assume $(E, e_0)$ and $(B, b_0)$ are based spaces with the indicated base points. Given a based fibration $p: E \rightarrow B$. We have the respective homotopy: fiber \begin{equation} Fp= ...
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1answer
52 views

Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
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22 views

Reference for introductory text on Homotopy Theory

Can anyone recommend a good introductory text on Homotopy Theory? Paid textbooks or free online material/lecture notes both welcome.
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50 views

A problem concern about the relationship between homotopy groups.

I encountered the following problem: Let $X$ be a closed manifold, $S^n$ be the $n$-dimensional standard sphere and denote $\Omega(S^n,X)$ be the space of base point-preserving maps from $S^n$ to ...
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2answers
74 views

What's the calculation formula of topological number for mappings of $\pi_{3}(S^2)=\mathbb{Z}$?

It is well-known that, when mapping $|\vec{n}(\vec{x})|=1$, we can use $N=\int{\mathrm{d}x_1\mathrm{d}x_2\vec{n}\cdot(\partial_1\vec{n}\times\partial_2\vec{n})}$ to calculate the topological winding ...
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1answer
60 views

Obstruction to reduction of structure group

In the wiki article it's stated that the obstruction to reduction of structure group along a morphsim $H \to G$ can be stated in terms of classifying spaces via the cofibre $BG/BH$ as follows. A ...
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1answer
38 views

About the Degree of a Map

I am reading Elements of Homotopy Theory by George W. Whitehead. In the section about maps of the $n$-sphere into itself, in the second last paragraph of the text quoted below, he says that "Then an ...
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Second Volume of Elements of Homotopy Theory?

In the preface of Elements of Homotopy Theory (GTM 61) by George W. Whitehead, he wrote that "I plan to devote a second volume to these developments". Does any one know if George eventually published ...
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26 views

What is “Triangulable Triad”?

I am reading George W. Whitehead's Homotopy Theory; Corollary 1.0.2 mentioned the term "Triangulable triad" without definition. May I know how it is defined?
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29 views

chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$ C^n(Y_i) \oplus C^{n-1}(X_i) $$ with boundary operator: $$ (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
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3answers
88 views

covering map $S^n \rightarrow P^n$ is not null homotopic

Here is the problem: Prove that the covering projection $S^n \rightarrow P^n$ is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ...
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1answer
32 views

CW complex with no cells in dimension $n$

Hi need some help with the following problem: if $X$ is a CW complex with no cells of dimension $n$ then $\tilde{H}^n(X,G)=0$. where $G$ is any group. thanx.
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1answer
27 views

Proving the left lifting property for a map

I want to prove the left lifting property for the inclusion map of the sphere into the disk for any fibration $q:X \rightarrow Y$, where $q$ is a weak equivalence. I don't know how to draw a square ...
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1answer
24 views

In the definition of $n$-equivalence, what is the motivation for only requiring surjectivity on the $n$th dimension.

An $n$ equivalence $f\colon X \to Y$ such that the induced map on the homotopy group $f_* \colon\pi_m(X) \to \pi_m(Y)$ is an isomorphism for $m<n$ and an epimorphism for $m=n$. What's the ...
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1answer
39 views

The unit ball in $\mathbb{R}^{n}$ and a point are homotopically equivalent.

I will appreciate if someone could explain to me the solution of the following problem: The unit ball in $\mathbb{R}^{n}$ and a point are homotopically equivalent. Def 1: Two spaces $X$, $Y$ are ...