All Questions
Tagged with kt.k-theory-and-homology at.algebraic-topology
306 questions
4
votes
2
answers
256
views
Waldhausen S-construction for exact categories
Let $\mathcal{C}$ be an exact category. Then, we can consider $\mathcal{C}$ as a Waldhausen category, where the cofibrations are admissible monomorphisms. By Waldhausen $S$-construction we know that $...
- 167
2
votes
0
answers
205
views
What role does homotopy play in Karoubi's K-Theory?
In Karoubi's book K-Theory An Introduction, he defines the groups $K^{p,q}(\mathcal{C})$ for a pseudo-abelian Banach category as equivalence classes of triples $(E,F,\alpha)$, where $E,F \in \mathcal{...
- 233
2
votes
1
answer
399
views
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 2,964
4
votes
1
answer
418
views
Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
- 41
6
votes
1
answer
186
views
Plus construction of the product spaces
I am newly learning plus construction in topology. My question is how to prove the following:
The plus construction of the product of two CW complexes is homotopically equivalent to the product of ...
- 613
1
vote
0
answers
111
views
Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
- 11
13
votes
1
answer
558
views
Intuitive reason for periods of 2 and 8 in Bott periodicity?
Is there a reasonably simple explanation for why Bott periodicity for $U$ and $O$ have periods 2 and 8, respectively? For example, in the $h$-cobordism theorem the requirement that $n \geq 5$ has the ...
- 131
6
votes
1
answer
426
views
Nilpotency of generalized cohomology
$\newcommand\pt{\mathrm{pt}}$Let $(X,\pt)$ be a connected, pointed, finite CW complex and let $h$ be a generalized cohomology theory. Let $\smash{\tilde{h}}^*(X)$ denote the kernel of restriction $h^*(...
- 959
3
votes
2
answers
246
views
Explicit description of transfer for $K_1$
Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence
$$
\ldots \rightarrow K_i(R/s) \rightarrow K_i(...
2
votes
0
answers
101
views
A roof genus of high dimensional lens space
Let $p$ be a natural number, and for $i\in \{0,
..., p-1\}$,
denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$.
Let $a=(a_{1},\ldots a_{d}) $ ...
- 2,630
7
votes
1
answer
101
views
Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces
From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...
- 71
0
votes
0
answers
86
views
Projectivity of equivariant K-theory of toric variety
I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups.
In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
- 959
13
votes
1
answer
2k
views
Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection
I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
- 155
3
votes
0
answers
90
views
When does homology preserve inverse limits of Eilenberg-MacLane spaces?
Let $... \to G_3 \to G_2 \to G_1$ be an inverse system of abelian groups and $G$ the limit of the system. By a theorem of Goerss the integral homology of the Eilenberg-MacLane space $K(G,n)$ for $n &...
- 499
13
votes
1
answer
385
views
Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?
There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...
- 56.9k
5
votes
0
answers
165
views
Equivalent descriptions of equivariant K-theory
I am looking at references for computing $$K_{T}(G/H)$$ where $G$ is a compact connected Lie group with maximal torus $T$, and $H\subset G$ is a corank one Lie subgroup such that $G/H\cong S^{2k-1}$ ...
- 51
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 ...
- 21
3
votes
0
answers
99
views
What does homotopy invariance mean for twisted K-theory?
In ordinary K-theory, homotopy invariance means that if $f,g \colon X \to Y$ are homotopic maps then their induced maps on K-theory are equal: $f^* = g^* \colon K(Y) \to K(X)$.
My question is how to ...
- 293
3
votes
1
answer
213
views
"High-dimensional" classes in topological $K$-theory
I am looking for a sequence of topological spaces $X_n$, $n\in\mathbb N$, with the following property. Let $\tilde{K}^0(X_n)$ be the complex reduced $K$-theory group of $X_n$ (with respect to some ...
- 1,903
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 ...
- 54.6k
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 ...
- 54.6k
5
votes
1
answer
468
views
The graded multiplication on topological $K$-theory
In every reference I have looked at (the books by Atiyah, Karoubi, Lawson--Michelsohn, Hatcher's unpublished book) the exterior multiplication on (reduced, negative) $K$-theory is given by the ...
- 18.2k
4
votes
1
answer
476
views
Equivariant K-theory for products of groups?
Let $X$ be a $(G \times H)$-space. What is known about the connection between the groups $K_G(X)$, $K_H(X)$ and $K_{G \times H}(X)$? The $G$ and $H$ action on $X$ come from the canonical inclusions $G ...
- 293
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 ...
user493302
6
votes
0
answers
128
views
K-homology fundamental class for singular varieties?
Given a smooth $\text{Spin}^c$ compact manifold without boundary $M$, a suitable normalization of the Dirac operator defines the fundamental class of $M$ in Kasparov's $KK(\mathbb{C}, C^0(M))$. This ...
- 915
5
votes
0
answers
129
views
Solving polynomial equations in $K(h)$-local or $T(h)$-local spectra?
This is the same question as an earlier question of mine, except in a different category.
Let $Spt_{T(h)}^{fin}$ be the category of finite $T(h)$-local spectra. Let $K_0^\oplus(Spt_{T(h)}^{fin})$ be ...
- 63.9k
5
votes
0
answers
161
views
Splitting of $BGL_1(KR)$
There are infinite loop space splittings $BGL_1(KO)\simeq BGL_1(KO)[0,2]\times Z$ and $BGL_1(KU)\simeq BGL_1(KU)[0,3]\times Z'$ where $Z$ and $Z'$ are 2 and 3 connected, respectively (i.e. they have ...
- 10.4k
4
votes
1
answer
457
views
Pullback of complex vector bundles along a retraction of compact Hausdorff spaces: a direct proof instead?
Consider a pointed compact Hausdorff space $(X,x_0)$ and a closed pointed subspace $i:A\subset X$ such that there exists a continuous map $r:X\rightarrow A$ such that $r|_A=\text{Id}_A$. Set
$$q:(X,...
user490645
6
votes
1
answer
379
views
The optimal ranges for the integral homological stability of $\operatorname{GL}_n(F)$'s for a field $F$
$\DeclareMathOperator\GL{GL}$
$\DeclareMathOperator\co{H}$
$\DeclareMathOperator\ko{K}$
$\DeclareMathOperator\trd{tr-deg}$
$\DeclareMathOperator{\ch}{char}$Given a field $F$ and a homological degree $...
- 1,726
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 ...
user484289
7
votes
1
answer
278
views
What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?
H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi:
...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...
- 609
13
votes
2
answers
586
views
When are bundles of odd and even differential forms isomorphic?
Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ ...
- 501
7
votes
1
answer
579
views
Topological K-theory of Riemann surface
Let $X$ be a compact Riemann surface of genus $g$, then $K^1_{\mathrm{top}}(X)\cong\mathbb{Z}^{2g}$. Is there a explicit description of a set of basis of $K^1_{\mathrm{top}}$? (e.g., For cohomology $H^...
user39380
8
votes
0
answers
440
views
Poincaré duality for topological $K$-theory
Let $n$ be an even number. Let $X$ be a $n$-dimensional complex projective manifold with
$H^{2m+1}(X,\mathbb{Z})=0$, for all $0\leq m\leq n-1$.
$H^{2m}(X,\mathbb{Z})$ is a free $\mathbb{Z}$-module ...
user39380
4
votes
1
answer
288
views
The third homology stability of general linear groups over finite fields
Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\...
- 441
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 ...
- 103
8
votes
0
answers
189
views
Conner-Floyd Chern classes and $E$-(co)homology of $BU$
In his book, Stable homotopy and generalised homology, Adams computes the $E$-(co)homology of $BU$ for a complex oriented cohomology theory $E$. In II.4, he first describes the $E$-homology of $BU$ as ...
user340793
21
votes
1
answer
837
views
What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?
Consider the following partial order. The objects are unordered tuples $\{V_1,\ldots,V_m\}$, where each $V_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V_1 \oplus \cdots \oplus V_m =...
- 1,025
4
votes
1
answer
617
views
Computation of KO theory of a point
I have some basic questions about real K-theory (I mean $KO$-theory).
Question 1: I have seen the table
$$
KO^{-i}(\mathrm{pt})=
\begin{cases}
\mathbb{Z},& i=0\\
\mathbb{Z}_2,& i=1\\
\mathbb{Z}...
- 1,903
21
votes
2
answers
1k
views
Does Waldhausen K-theory detect homotopy type?
Recall that $A(X)$, the K-theory of a connected, pointed space X, is defined as the K-theory spectrum of the ring spectrum $\Sigma^\infty_+ \Omega X$ (or via a plethora of alternative definitions). Is ...
- 5,839
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 ...
- 7,637
5
votes
2
answers
442
views
Sheaf of chain complexs glued by chain homotopy equivalences
Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose:
For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-...
- 2,789
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 ...
11
votes
1
answer
520
views
Problems concerning subspaces of $M_{n}(\mathbb{Q}) $
Let $M_{n}(\mathbb{Q}) $ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M_{n}(\mathbb{Q}) $.Then if all the non-zero matrices in $N$ are invertible, what is ...
- 923
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 ...
- 101
6
votes
1
answer
426
views
What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?
The cohomology ring of $\mathrm{SL}_2(\mathbb C)$ as a topological group is straightforward (it's generated by a Chern class), but what is known in the discrete case? I'm particularly interested in $H^...
- 2,533
3
votes
0
answers
218
views
What is the multiplicative structure of K-theory $\Omega$-spectrum $KU$?
The K-theory $\Omega$-prespectrum $KU$ has spaces $KU_{2i}=BU\times \mathbb{Z}$ and $KU_{2i+1}=U$, according to Bott periodicity we have a $\mathbb{Z}_2$-graded cohomology theory $\tilde{K}^*(X)$, ...
- 71
5
votes
1
answer
259
views
Integral homology of braid groups as a ring
Let $Br_k$ denote the braid group on $k$ strands. In Corollary A.4 of "Homology of Iterated Loop Spaces" (Page 348), Cohen-Lada-May compute $H_i(Br_k;\mathbb Z)$ as an abelian group for each ...
- 965
19
votes
1
answer
862
views
Diffeomorphism groups of h-cobordant manifolds
Do we have specific examples of h-cobordant smooth manifolds $M$ and $M'$ such that $\operatorname{BDiff}(M) \not \simeq \operatorname{BDiff}(M')$? Perhaps something can be said in terms of K-theory ...
- 5,839
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 ...
- 63.9k