Questions tagged [stable-homotopy-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
0answers
17 views

Inflation of Equivariant Eilenberg-Mac Lane spectra

Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Set $Q=G/N$ -- the quotient group. Consider the quotient homomorphism $\pi: G \to G/N$. It induces a functor $\pi^\ast\colon Q\text{-...
2
votes
1answer
74 views

$E^*(\mathbb{C}P^{\infty})=\bigoplus_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$ or $\prod_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$?

PRELIMINARY DEFINITIONS: Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have: $$ \tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt) $$ So there is a special ...
4
votes
0answers
64 views

Are there finitely many trivial stable stems?

One can look up in a table that for example $\pi_4^s = \pi_5^s = 0$. However, it seems to be that the stable homotopy groups of spheres get larger and larger for higher dimensions. Question: is it ...
1
vote
1answer
18 views

Degree of a pmap

Today I have started reading stable homotopy. I have came across the notion of a pmap which is basically equivalence class of maps from cofinal sub-spectra. My query is what do we mean by degree of ...
0
votes
0answers
59 views

Stable homotopy groups as a generalized (reduced) homology theory

It is known that $\pi_*^{st}$ defines a generalized homology theory, where $\pi_n^{st}(X) = \text{colim}_{k \geq 0} \pi_{n+k}(\Sigma^k X)$ is the $n$th stable homotopy group of the based space $X$. ...
0
votes
0answers
35 views

Representation Theory and Equivariant Stable Homotopy Theory

What is a good book/source to understand Representations of a Group in the sense we use it in Equivariant Stable Homotopy Theory? I've read Barry Simon's book on Representation Theory but would like ...
3
votes
0answers
76 views

Which (co)homology theories have (co)chain complexes, spectrum edition?

What homological functors $\mathbf{Sp}\rightarrow\mathbf{Ab}$ arise in the form $H_0\circ T$, for $T:\mathbf{Sp}\rightarrow D(\mathbf{Ab})$ a triangulated functor, and dually for cohomological ...
3
votes
0answers
82 views

Mahowald-Hopkins theorem

I am trying to recover fully a proof of a result which is attributed to M. Hopkins in the litterature, namely the following: Let $h:S^1\to BGL_1(\mathbb S_p)$ detect the element $1-p$, then the Thom ...
2
votes
1answer
44 views

Why is the Moore spectrum $S\mathbf{Z}_{(J)}$ a ring spectrum?

Let $J$ be a set of primes and consider the Moore spectrum $S\mathbf{Z}_{(J)}$. In his paper 'The localization of spectra with respect to homology', Bousfield writes that $S\mathbf{Z}_{(J)}$ is a ring ...
1
vote
0answers
52 views

A non modern theory of generalized Thom spectra

I've already posted this question on mathoverflow a few days ago and I had no reaction, I hope it is not illegal to repost it on math.stackexchange. I'm new in the subject of stable homotopy theory, ...
1
vote
0answers
37 views

The infinite loop space associated to the spectrum associated to a special $\Gamma$-space

I am currently reading M. Mandel's paper `An Inverse $K$-theory Functor' to extract some results I may wish to use in my masters research project, and I am stuck on one particular claim. On the top of ...
0
votes
0answers
35 views

Cofinal functor to category of orthogonal $G$ representations.

Let $G$ be a finite group. $R_G$ denote category of finite dimensional orthogonal $G$-representation. (i,e. f.d. real vector spaces with inner product compatible with action of $G$), It is claimed ...
5
votes
0answers
122 views

$G$-spaces vs spaces with a $G$-action

In equivariant homotopy theory, it seems like one tends to consider "genuine" $G$-spaces or $G$-spectra, rather than spaces (spectra) with a $G$-action. My (rather soft) question is : why is that a ...
3
votes
0answers
43 views

Postnikov sections are monoidal functor or not?

For any space/spectrum $X$ one can define the associated tower of Postnikov sections $\{ P^n X\}_{n}$ as a Bousfield localization with respect to all spheres with dimension $>n.$ Therefore, we have ...
1
vote
0answers
54 views

What does it mean for this proposition to hold in $Top_*$?

In chapter 2 (or 1 depending on your edition) of their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim have a certain Proposition 2.1.9 (or 1.1.9) saying that if $A\to X$ is an h-...
1
vote
0answers
38 views

Symmetric spectrum associated to a ring

In this paper by Dwyer, Greenlees and Iyengar, we are introduced to symmetric spectra, and more particularly to the notion of an $\mathbb{S}$-algebra. In both the Notation and terminology ($1.5$), ...
2
votes
0answers
65 views

Why the category of CW-complexes is not stable?

I am studying homotopy theory and I would like to understand better what it means for a category to be stable. For instance, the book I'm studying says that "the categories CW∗ and CCh+ have very ...
1
vote
0answers
61 views

CW Wedge sum Cofiber sequence

I was reading this paper by Bousfield on the localization of spectra. On page 5, Lemma 1.13, there's a rather small curious technical detail on wedge sum. We have for a limit ordinal $\lambda,B_{\...
0
votes
1answer
45 views

How do I prove that all smooth manifolds being homotopic implies contractible?

Let $X$ and $Y$ be smooth manifolds. If all smooth maps from $Y$ to $X$ are homotopic, then show that the identity map on $X$ is homotopic to some constant map(i.e that $X$ is contractible). I have ...
1
vote
1answer
49 views

Homology of a spectrum

Let $X$ be a spectrum and $E$ another spectrum (it'll be our coefficients, if it makes things easier I'm ok woth assuming $E=H\mathbb Z$) The definition of $E_nX$ is usually given as $\pi_n(E\wedge X)...
4
votes
0answers
63 views

$KO_*$ groups of $\mathbb{R}P^\infty$, “Snaiths” theorem for $KO$

I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The standard tools to compute these Groups in the complex case rest on the requirement for the cohomology theories $E$...
2
votes
1answer
40 views

Multiple products on $R^* R$ for a ring spectrum R

Suppose we have a ring spectrum R with multiplication $\mu$ that also has a diagonal map $\Delta$ (for example the sphere spectrum). The cohomology $R^* R$ has a ring structure, like it would for any ...
3
votes
0answers
41 views

Notion of stability for Lusternik-Schnirelman category

I was browsing through survey on Lusternik-Schnirelman category and I became curious is it possible to give the definition of the category using a more invariant approach than the classical definition?...
2
votes
0answers
22 views

Stable homotopy of exceptional Lie groups

Using Bott periodicity, we know the stable homotopy of all classical compact lie group. So I am wondering if a similar pattern exists for exceptional Lie groups. Unfortunately, I couldn't find a ...
1
vote
1answer
29 views

Field spectra and Eilenberg--MacLane spectra?

Apparently thanks to a theorem of Hopkins and Smith, every field spectrum splits into a wedge of Morava K-theories, where we allow the cases $K(0) = H \mathbb{Q}$ and $K(\infty) = H \mathbb{F}_p$. I ...
2
votes
1answer
80 views

Confusion about Phantom Maps

A phantom cohomology operation (originally read phantom map) $f: X \rightarrow Y$ is a non-nullhomotopic map such that the induced cohomology operation on the cohomology theories for spaces is trivial....
2
votes
1answer
55 views

Why is $E^*(X)$ graded commutative?

Context: I'm reading The stable homotopy category by Malkiewich (see his webpage, at Expository writings), and at this point the author hasn't actually introduced spectra, just the properties we would ...
4
votes
0answers
59 views

Additivity in algebraic K-theory — what does it truly mean?

--- Question --- I have seen several definitions of 'additivity' in algebraic K-theory. In all cases, I can more or less see that there is something additive going on. But I have difficulty seeing ...
2
votes
1answer
48 views

Spectra and topological diagrams, nlab

I am referring to the long Proposition 1.23 of U. Schreiber's notes in nlab. We let $X$ be a functor from $StdSphere\rightarrow Top^{*/}_{cg}$. He states that there is a map , where $X_i^{seq}=X(S^i)$...
2
votes
1answer
52 views

Uniqueness of a homotopy category.

For a category with weak equivalences $(C,W)$ call $(ho(C),F)$ a homotopy category of $(C,W)$ where $ho(C)$ is a category and $Q \in Fun^{W}(C,ho(C))$ is a functor inverting $W$ if the following ...
2
votes
0answers
67 views

Functoriality of twisted K-theory

In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed until now. I'm trying to use May-Sigurdsson's definition of Twisted $K$-theory, ...
1
vote
0answers
37 views

Definition of coevaluation map of a stable category defined in Tammo tom Dieck's Algebraic Topology.

The following picture is from [Tammo tom Dieck, Algebraic Topology, pp.176]. Question I want to ask if the statement that $j$ induces an h-equivalence : $$C(V|K)\overset{}{\longrightarrow}C(\mathbb{...
0
votes
1answer
391 views

Smash product of $S^1$ with the interval $I$

I'm trying to work through various examples of smash products of spaces. In order to check to see if what I'm doing is correct, is the smash product $S^1 \wedge I$ of the circle with the interval $[-1,...
1
vote
0answers
30 views

Values of the $\mathbb{Z}/2$-spectrum $KR$ of $K$-theory with Reality

Atiyah's $K$-theory with Reality produces a $\mathbb{Z}/2$-spectrum $KR$. But I am stuck as I don't really know what the values of this equivariant spectrum should be on representations. Any ...
4
votes
0answers
69 views

Does an $X$ exist so $X \simeq \Omega^n Y$ for all $n$, and X is not an infinite loop space?

Are there examples of spaces $X$ that have the property $X \simeq \Omega ^n Y$ with $Y$ depending on $n$, for all natural $n$, with $X$ not an infinite loop space? I have a feeling these things ...
2
votes
0answers
53 views

Commutativity of monoid object on $\mathcal{D}$-spaces

Let ($\mathcal{D}, \otimes, 0)$ be a symmetric monoidal category enriched over pointed spaces, and denote by $\mathsf{Top}_*^\mathcal{D}$ to the category of "$\mathcal{D}$-spaces", the category of ...
3
votes
0answers
65 views

Filtered colimits in Adams' category

I am currently reading Part 3 of Adams's book "stable homotopy and generalized cohomology", and I got stuck when following his argument. In Proposition 5.4, he states that when $W$ and $X$ are finite ...
0
votes
1answer
77 views

Reference for the cohomology of SU

Let SU be the infinite special group. Where can I find the following fact (state in part III of the Adam's blue book): $H^{6}(SU,Z)=0$. Thank you.
1
vote
1answer
55 views

Factorization of the orientation map $MU\to H\mathbb{Z}$ through $ku$?

Let $MU$ denote the complex cobordism spectrum and $ku$ the connective cover of the complex $K$-theory spectrum. Is it true that the orientation map $MU\to H\mathbb{Z}$ factors through $ku$?
3
votes
1answer
103 views

Detecting $\eta^3$ in stunted projective spaces.

Consider the stunted complex projective space $\mathbb{C}P^{n+2}_n:=\mathbb{C}P^{n+2}/\mathbb{C}P^{n-1}$ which is a three-cell complex of the form $$\mathbb{C}P^{n+2}_n\simeq S^{2n}\cup_{\alpha_n} e^...
1
vote
1answer
404 views

Connections between Algebraic Topology and Set Theory

(Co) Homology functors are dependent on the homotopy type of the objects they act on and so a lot of results only care about the "loose" classification of spaces (including the use of co-final spectra ...
1
vote
0answers
74 views

Tate construction of the Anderson dual of the sphere spectrum

I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
2
votes
1answer
99 views

Mod p cohomology operations and homotopy groups of spheres

It is well-known that "the $p$-torsion in the stable homotopy groups of spheres originate in $\pi_{2p}(S^3)$, but how can I prove this for odd primes? My idea would be to show that the mod $p$-...
0
votes
1answer
75 views

Is the complex cobordism spectrum, $MU$, a finite spectrum?

Is the complex cobordism spectrum, $MU$, a finite spectrum? If yes, what other examples of finite spectrums there are? Is the Eilenberg-MacLane spectrum finite? What about the connective $K$-theory $...
0
votes
1answer
122 views

Spectra and long exact sequences

Suppose $X,Y,Z$ are finite CW complexes and let $f \colon X \to Y$ be a fibration with fiber $F$. Is there now a long exact sequence associated to their suspension spectra in the following sense $$ \...
9
votes
1answer
278 views

Are there non-contractible space such that suspension $\sigma\colon \pi_k(X) \to \pi_{k+1}(\Sigma X)$ is an isomorphism for all $k$?

Does there exist a path-connected non-contractible CW-complex $X$ such that suspension $\sigma\colon \pi_k(X)\to \pi_{k+1}(\Sigma X)$ is an isomorphism for all $k$? If so is there also a simply-...
5
votes
0answers
164 views

Spectra and cohomology theories

I know that every generalised (Eilenberg-Steenrod) cohomology theory defines a spectrum (in the sense of Lewis-May), and vice-versa. I also know that maps between spectra are richer than maps between ...
3
votes
1answer
494 views

Book on stable homotopy theory?

Currently I know nothing about stable homotopy theory other than that it originated from the Freudenthal suspension theorem. But I believe that the following are studied in this field: spectrum, ...
1
vote
1answer
79 views

How are multiplication maps of spectra defined?

In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition. For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} \mathrm{MU}(p) \to \...
1
vote
0answers
21 views

Geometric proof of the fact that product of two elements of odd order in image of $J$ is zero.

I want to prove that product of two elements of odd order in image of $J$ is zero. I tried to approach this via the Thom-Pontryagin condition. So the image of $J$ just corresponds to spheres with ...