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equivalence of n-connected spectra

Is the following claim true in general? Claim: A map $f:(X^r)\rightarrow (Y^r)$ of $n$-connective spectra (of simplicial sets) is a stable equivalence if and only if the corresponding map ...
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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 ...
3
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0answers
41 views

“Naive” smash products for spectra

Suppose I work in the completeley naive homotopy category of spectra, by which I mean sequences $E = (E_n)_{n = 0, 1, \dots}$ together with maps $\sigma_{E,n}: S^1 \wedge E_n \to E_{n+1}.$ We might ...
2
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1answer
122 views

Refining homotopy commutative maps of spectra to maps of E_{\infty}-ring spectra

In Adams' blue book (page 54) we have a map in the homotopy category of ring spectra $f: MU \rightarrow K$ where $K$ is complex $K$-theory such that $g_*x^{MU} = (u^K)^{-1}x^K$ where $x^E$ denote ...
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1answer
56 views

Why is a spectrum $X$ with only $\pi_0X\neq 0$ equivalent to an Eilenberg-MacLane spectrum?

For an abelian group $G$, the Eilenberg-MacLane spaces $K(G,k)$ assemble to a spectrum $HG$ with $HG_k=K(G,k)$. This spectrum has the property that $\pi_0 HG=G$ and $\pi_{n}HG=0$ for $n\neq 0$ where ...
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2answers
206 views

Why does the loopspace $\Omega$ induces a weak equivalence on mapping telescopes?

I am trying to answer an exercise of Hatcher's "Algebraic Topology", Section $4$.F, exercise $3$. Suppose we are given a sequence of pointed topological spaces : $Z_0\rightarrow Z_1\rightarrow Z_2 ...
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0answers
125 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
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1answer
72 views

Map induced by $O(n)\hookrightarrow U(n)$ on homotopy groups

There is an inclusion $O(n)\hookrightarrow U(n)$ which views an $n\times n$ orthogonal matrix as a unitary matrix. It is also a theorem, sometimes called Bott periodicity, that we have the following ...
3
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1answer
112 views

Rational spectra

I keep reading the term "rational (ring-)spectrum" but can't find a definition. My original motivation for researching this was to understand some basic examples of complex oriented cohomology ...
1
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1answer
151 views

Loop spaces and filtered colimits

I have read many things now that lead me to believe that the loop space functor preserves filtered (and/or directed) colimits. Is this true? And can somebody give a (sketch of a) proof or point me in ...
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0answers
63 views

Hurewicz isomorphism in equivariant stable homotopy

Let $G$ be a finite group and let $X$ be a $G$-CW-complex. Denote by $\pi_{\ast}^G(X)$ the $G$-equivariant stable homotopy groups of $G$ and by $\mathrm{H}_{\ast}^G(X,A(-))$ the Bredon homology of $G$ ...
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41 views

Computing group of exotic spheres

In Levine on page 90 it is stated that the following sequence is exact $$ 0 \to bP^{n+1} \to \Theta^n \to Coker(J_n) $$ where $\Theta^n$ is the group of exotic spheres, $bP^{n+1}$ is the subgroup of ...
2
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0answers
63 views

What happens in dimension 125?

In Differential topology 46 years later (page 807, bottom of left column) Milnor states that for $n \neq 4, 125, 126$ if the order of the stable homotopy groups $|\Pi_n|$ is known then we can compute ...
5
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1answer
145 views

What is the $p$-primary component?

I got stuck when reading Differential topology 46 years later in the last section of the article ("Further details"). It is a summary of what is known about stable homotopy groups of spheres $\Pi_n$. ...
2
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1answer
76 views

Bott periodicity and homotopy groups of spheres

I studied Bott periodicity theorem for unitary group $U(n)$ and ortogonl group O$(n)$ using Milnor's book "Morse Theory". Is there a method, using this theorem, to calculate $\pi_{k}(S^{n})$? (For ...
4
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1answer
84 views

Different point-set level definitions of spectra

I've been trying to understand the Adams spectral sequence and one of the more accessible sources is the (unfinished) book on spectral sequences by Hatcher. The usual (only?) construction of the ...
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2answers
360 views

Motivation of stable homotopy theory

A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take $$\lbrace X,Y \rbrace = ...
2
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2answers
213 views

(weak) homotopy equivalence

I have a question arising from chapter 3, page 41, in Switzer. He says "Note that every homotopy equivalence (in $\mathscr{T}$ [this is the category of topological spaces]) is a weak homotopy ...
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3answers
251 views

What's the point of spectra?

I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
1
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1answer
79 views

Question regarding the reduced suspension functor

There is a step in J.F.Adams book, Infinite Loop Spaces, which I don't quite understand. Here is the whole extract: Let $W$ be a further space (not sure what 'further' means, seems unnecessary), ...
2
votes
1answer
92 views

Different possibilities defining $\eta^2$ in the ring of stable homotopy groups?

The Hopf fibration $\eta:S^3\to S^2$ represents the generator of the first stable homotopy group $\pi_1^s$. The direct sum of the stable homotopy groups $$ \pi_*^s=\bigoplus_{k\in\mathbb{Z}} \pi_k^s ...
2
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2answers
46 views

Solving Recursions Without Initial Conditions

I am trying to solve the following recursion but it does not appear that I can use characteristic equations or generating functions since I do not have initial conditions. Is there another way I am ...
0
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1answer
129 views

Closed orbits of SIR model variants

I've experimented with a few different SI and SIR models in an attempt to find some closed orbits. So far, I've had no luck. (Note that $\beta$ is the infection probability/rate, $b$ is constant ...
3
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1answer
53 views

Stable homotopy groups of $Sp(2n)$

What are the stable homotopy groups of the symplectic groups $Sp(2n)$? Is there a reference which contains a detailed treatment?
1
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1answer
56 views

stable homotopy groups zero

A connected CW-complex $X$ is contractible iff the homotopy groups $\pi_n(X)$ are zero for all $n\geq 1$. What (if any?) is the analogous statement for the vanishing of all stable homotopy groups ...
2
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0answers
108 views

Massey products in the Adams Spectral Sequence

I've never quite 'got' Massey products - this question, I guess, is to work out a small example that might shed some light for me. So following Wikipedia, let $\Gamma$ be a differential graded ...
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0answers
52 views

$\mathbb{S}[q^{-1}]$ is Moore spectrum $M(\mathbb{Z}[q^{-1}],0)$?

Let $S[q^{-1}]$ denote the homotopy colimit of the sequence $S\overset{q}\to S\overset{q}\to\cdots$ where $S$ denotes the sphere spectrum. Is it then the case that $S[q^{-1}]$ is the Moore spectrum of ...