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51 votes
5 answers
5k views

What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra. In the former, one assigns to ...
Dan Petersen's user avatar
  • 40.2k
39 votes
3 answers
2k views

Lambda-operations on stable homotopy groups of spheres

The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable ...
skupers's user avatar
  • 8,167
31 votes
2 answers
3k views

Conceptual explanation for the relationship between Clifford algebras and KO

Recall the following table of Clifford algebras: $$\begin{array}{ccc} n & Cl_n & M_n/i^{*}M_{n+1}\\ 1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\ 2 & \mathbb{H} & \mathbb{Z}/2\...
Callan McGill's user avatar
30 votes
1 answer
2k views

Morava K-theories for dummies?

Professor Urs Würgler passed away one year ago, and his wife engraved his tombstone with "the formula he was the most proud of" : $B(n)_*(X)\cong P(n)_*(K(n))\square_{\Sigma_n}K(n)_*(X)$ However ...
Dr. Goulu's user avatar
  • 403
26 votes
1 answer
2k views

What is to tmf as KR is to KO?

The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...
Urs Schreiber's user avatar
17 votes
3 answers
3k views

Finiteness of stable homotopy groups of spheres

Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or ...
Andreas Holmstrom's user avatar
14 votes
2 answers
715 views

The $K$-theory homology of the Eilenberg-MacLane spectrum

Let $KU$ be the complex $K$-theory spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. For $n\in \mathbb{Z}$, it is known what the homology groups $KU_{n}(H\mathbb{Z})$ are?
Tsk's user avatar
  • 578
13 votes
1 answer
385 views

Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?

There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...
Neil Strickland's user avatar
11 votes
4 answers
1k views

Maps from mod-$p$ Eilenberg-MacLane spectrum to connective $K$-theory spectrum

Let $ku$ be the connective cover of the complex $K$-theory spectrum $KU$. Consider the mod-$p$ Eilenberg-MacLane spectrum $H\mathbb{Z}/p$. I want to see that $[H\mathbb{Z}/p,ku]=0$. Since $H\mathbb{...
user438991's user avatar
11 votes
2 answers
864 views

Solving polynomial equations in spectra?

Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general ...
Tim Campion's user avatar
  • 63.9k
11 votes
0 answers
532 views

Chromatic Homotopy Theory and Physics

Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
wonderich's user avatar
  • 10.5k
10 votes
1 answer
332 views

Which spectra have a universal connective quotient?

Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group ...
Theo Johnson-Freyd's user avatar
10 votes
0 answers
325 views

Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$

First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
Francis Baer's user avatar
9 votes
1 answer
2k views

Dennis trace map K----> THH

I have some questions about Dennis trace map in algebraic K-Theory. I was wondering if there is some conceptual way to look at this map $K(-)\rightarrow THH(-)$ (natural transformation from K-Theory ...
Ilias A.'s user avatar
  • 1,974
9 votes
0 answers
371 views

Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?

First of all, what I want to ask is slightly more elaborate than what stands in the title (hence the subquestion). I am telling this since as it is, the title contains a meaningful question, but it ...
მამუკა ჯიბლაძე's user avatar
8 votes
1 answer
271 views

$K$-theory and its dual

I am reading a paper which uses some $K$-homology which is the homology theory dual to $K$-theory can be defined using the homotopy theoretic formulation: $$ K_\ast(X)\cong\pi_\ast(K\wedge X). $$ ...
Louis T's user avatar
  • 81
7 votes
1 answer
414 views

Reference request: mod 2 cohomology of periodic KO theory

The mod 2 cohomology of the connective ko spectrum is known to be the module $\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$, where $\mathcal{A}$ denotes the Steenrod algebra, and $...
Nicolas Boerger's user avatar
7 votes
1 answer
410 views

Does a Gysin map depend on the choice of Thom class?

Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom ...
user avatar
7 votes
1 answer
274 views

Proof of the equivalence of spectra $(\mathbb{S}^{-1} \otimes \mathbb{S}^{-1})_{h \Sigma_2} \cong \Sigma^{-1} \mathbb{RP}_{-1}^{\infty}$

$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\Th}{Th}$I am trying to give a hands-on proof of the equivalence of spectra in the title. I am using the definitions $\mathbb{RP}^{\infty}_{-...
Steven's user avatar
  • 348
7 votes
1 answer
462 views

Cohomology theory with only one Adams operation?

Let $E$ be a multiplicative cohomology theory. Fix a prime p. Call a ring map $\psi^{p}:E\rightarrow E$ an Adams operation if it lifts the Frobenius map $E/p\rightarrow E/p$. It is of course well-...
John Greenwood's user avatar
7 votes
0 answers
265 views

Connecting Harris's proof of periodicity to the Bott element

$\newcommand\ku{\mathrm{ku}}$In Denis Nardin's lectures on stable homotopy theory, he reproduces Bruno Harris's proof of Bott periodicity from group completion and goes on to identify the Bott element ...
daniel gratzer's user avatar
7 votes
0 answers
327 views

Funtoriality of twisted K-theory

I posted this question on math.stackexchange, but received no answer there. 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 ...
Excalibur's user avatar
  • 301
7 votes
0 answers
344 views

Reference request: complex K-theory as a commutative ring spectrum

Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum? For real $K$-theory I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...
David Barnes's user avatar
6 votes
1 answer
401 views

Is ku reflexive as a spectrum?

Let $S$ and $\rm ku$ denote the sphere spectrum and the connective K-theory spectrum, respectively. Then is the morphism ${\rm ku} \to F(F({\rm ku},S),S)$ an equivalence? Here $F$ denotes the mapping ...
B. W.'s user avatar
  • 368
6 votes
1 answer
534 views

Stable Adams operations

I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not ...
Avishkar Rajeshirke's user avatar
5 votes
1 answer
345 views

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

I posted this question some days ago at math.stackexchange, but didn't receive an answer. I have two questions: I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The ...
Excalibur's user avatar
  • 301
5 votes
0 answers
173 views

Uniqueness of complex topological $K$-theory as an $S$-algebra

This might be well-known or trivial, but I could not figure out how to fill in the details: For an $S$-algebra $K$ denote its associated multiplicative cohomology theory by $h^*_K$. Suppose that I ...
Ulrich Pennig's user avatar
4 votes
1 answer
165 views

$E^G_\ast(E)$ tensored with the rationals

Lemma 17.19 of Switzer's "Algebraic topology - Homology and Homotopy" states that $E_\ast(F)\otimes\mathbb{Q}$ is isomorphic to $\pi_\ast(E)\otimes\pi_\ast(F)\otimes\mathbb{Q}$. I wanted to ...
user avatar
4 votes
1 answer
172 views

The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
atinag's user avatar
  • 43
4 votes
1 answer
282 views

Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory?

I am reading Ravenel's Localization with Respect to Certain Periodic Homology Theories where he states; For $n\ge2$, the spectra E(n) represent periodic homology theories which at present have ...
Gabriel Angelini-Knoll's user avatar
4 votes
1 answer
173 views

Injectivity of assembly in A-theory for $BC_2 = \mathbb R P^\infty$ in degree $4$

I am trying to understand the assembly map $$\pi_i ((BC_2)_+ \wedge A( \ast )) \rightarrow A_i( BC_2 ) $$ in low degrees for the space $BC_2 = \mathbb R P^\infty$ in Waldhausen $A$-theory. I know we ...
Georg Lehner's user avatar
  • 2,303
4 votes
0 answers
87 views

Shearing maps on domain of assembly map in algebraic $K$-theory

Let $H \to G$ be an inclusion of abelian groups, and let $R$ be a ${\Bbb Z}[H]$-algebra. Assume that the assembly map ${\Bbb S}[BG] \otimes_{\Bbb S} K(R \otimes_{{\Bbb Z}[H]} {\Bbb Z}[G]) \to K((R \...
Somewhat confused's user avatar
4 votes
0 answers
226 views

How to to understand the homology groups $H_*(\Omega_0^\infty S^\infty)$?

The original statement of the Barratt--Priddy theorem says there is an isomorphism of homology groups $$H_*(\Sigma_\infty)\cong H_*(\Omega_0^\infty S^\infty),$$ where $\Omega_0^\infty S^\infty$ is the ...
Chase's user avatar
  • 103
3 votes
0 answers
129 views

Which spectra have a homotopy-universal connective quotient?

Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the ...
Theo Johnson-Freyd's user avatar
3 votes
0 answers
187 views

$1$-periodic mod-$2$ K-theory

Complex $K$-theory mod $2$ is $2$-periodic, $K/2_* = \mathbf{F}_2[u,u^{-1}]$. Is there an extension $K/2 \to K'$ of ring spectra such that $K'_*=\mathbb{F}_2[q,q^{-1}]$ with $|q|=1$ and such that the ...
Tilman's user avatar
  • 6,162
2 votes
2 answers
275 views

The complex $K$-theory of the Thom spectrum $MU$

The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...
Plius's user avatar
  • 21
2 votes
1 answer
307 views

Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory

Let $X$ be an scheme. Fix a prime $l$ which is invertible in $X$. Consider the $K(1)$-localization at prime $l$ of algebraic K theory $L_1K(X)$ and $l$-adic completion of etale K theory $K^{et}(X)$. ...
Fredy's user avatar
  • 127
1 vote
0 answers
68 views

Techniques for Showing Triviality of K_1 of a Higher Category

Suppose $\cal{C}$ is a small stable $\infty$-category. Then, we have its K-theory spectrum $K(\cal{C})$ that gives us K-theory groups $K_n(\cal{C})$ by taking stable homotopy groups. There are ...
user110215's user avatar