5
votes
1answer
206 views

Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
14
votes
0answers
317 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 ...
5
votes
1answer
196 views

Equivalent fomulations of Bott periodicity

Is there an easy way to see the equivalence of the two statements of Bott periodicity. $$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and $$K(X)\otimes K(S^2) \cong K(X\times S^2)$$
13
votes
1answer
449 views

Adams' theorems on the Hopf-Whitehead J-homomorphism

The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$, a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of ...
1
vote
0answers
95 views

Is the $K$-Theory of $\mathbb{P}X$ a $\lambda$-Ring?

If we let $\mathbb{P}X$ denote the free commutative algebra generated by the spectrum $X$, then we have a weak equivalence $$ \mathbb{P}X\simeq \bigvee_r E\Sigma_{r+}\wedge_{\Sigma_r}X^{\wedge r}. $$ ...
4
votes
1answer
128 views

Yoneda embeddings of stable model categories; composition with Bousfield localizations

For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question ...
2
votes
2answers
296 views

filtration in K-theory and ordinary cohomology

I am going to ask a question, which could be a stupid one. I am reading a paper "an index theorem in differential K-theory". The first paragraph of section 8.28 recalls a filtration of K-theory ...
4
votes
0answers
114 views

Interpretation of the product $K(X)\otimes K^{-1}(X) \to K^{-1}(X)$

We can represent every element of the group $K^{-1}(X)=\tilde{K}(SX)$ by a isomorphism of trivial vector bundles $L:\, X\times \mathbb{C}^k \to X\times \mathbb{C}^k$ because $SX$ is the union of two ...
2
votes
0answers
161 views

Surfaces and the boundary map of K-theory

In the reduced K-theory we have the exact sequence of pair $$ \tilde{K}(S(X/A)) \to \tilde{K}(SX) \to \tilde{K}(SA) \to \tilde{K}(X/A) \to \tilde{K}(X) \to \tilde{K}(A) $$ If choose $X$ to be a ...
1
vote
1answer
207 views

Definition of the homological Chern character

There is a homological Chern character $ch_\ast \colon K_\ast(X) \to H_\ast(X)$ for $X$ a smooth, compact manifold. I found only one definition of it (in the paper "K-Homology and Index Theory" by ...
5
votes
1answer
179 views

What is the ring structure of the complex topological K-theory of a non-singular complex quadric?

I would like to know the ring structure of $K(Q_n)$ explicitly where $Q_n \subset \mathbb{P}^{n+1}$ is the non-singular $n$-dimensional complex quadric and $K(Q_n) = K^0(Q_n)$ is the complex ...
33
votes
2answers
2k views

What arithmetic information is contained in the algebraic K-theory of the integers

I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
4
votes
1answer
348 views

Representing KO-theory using Clifford algebras

I'm trying to understand a statement Segal makes in this book: Let $C_q$ be the real Clifford algebra associated to the standard negative definite form on $\mathbb{R^q}$ and let $\Phi_q(n)$ be the ...
6
votes
1answer
254 views

Coefficients of real k-theory with coefficients

Question: Calculate the group $ \pi_{8k+2}(KO \wedge M\mathbb Z/l\mathbb Z) $. Here $KO$ denotes the real k-theory spectrum and $M\mathbb Z/l\mathbb Z $ denotes a Moore Spectrum associated to the ...
13
votes
0answers
218 views

Is there an effective way to calculate K-theory using Morse functions?

Let $M$ be a compact manifold and let $f$ be a Morse function with exactly one critical point at each critical level. Then one can recover a CW-complex with the homotopy type of $M$ from just the ...
16
votes
0answers
387 views

Computational complexity of topological K-theory

I am a novice with K-theory trying to understand what is and what is not possible. Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...
7
votes
1answer
226 views

Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the ...
10
votes
3answers
638 views

Plus construction considerations.

In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction: Recall that $K_1(R) = ...
3
votes
1answer
228 views

A theory of bifurcation of braids ?

I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional ...
0
votes
2answers
302 views

The First Homology Group of Configuration Space and Knot Theory

Let $\pi_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the fundamental group functor and let $H_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the first homology group functor. We can then define ...
0
votes
2answers
114 views

free complex with mod-p coefficients

How does one prove the following fact. I could not find anything in literature. Let $\pi$ be a subgroup of the symmetric group $S_p$ and let $W$ be a free $\pi$-complex. Then for any space $X$ there ...
32
votes
3answers
736 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 ...
2
votes
2answers
251 views

Equivariant K-theory of $S^1$-action on $S^2$

Is there any references for the structure of the equivariant K-theory $K_{S^1}(S^2)$ where the action of $S^1$ on $S^2$ is defined to be rotation about the $z$-axis? What is the ring structore of ...
2
votes
0answers
136 views

Two questions on axiomatic homology

1) Given the Eilenberg-Steenrod axioms, there are several Mayer-Vietoris type sequences that can be deduced. The most general form seems to be $$\rightarrow H_n(X \cap Y, A \cap B) \rightarrow ...
21
votes
1answer
698 views

Is every ''group-completion'' map an acyclic map?

I start with a longer discussion which will result in a precise version of the question. A am puzzled about an issue with the Quillen plus construction. I have seen outstanding experts being confused ...
6
votes
0answers
494 views

Is there any “deep” relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex ...
12
votes
2answers
794 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 ...
12
votes
1answer
674 views

finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from algebraic ...
4
votes
1answer
181 views

What is the role of $\sum (-1)^p[\wedge^pT^*M]$ in the K-theory $K(M)$

I apologize for the vague title. Let $M$ be a compact smooth manifold, then we have $T^*M$ and hence $\wedge^pT^*M$ as vector bundles on $M$. There for we have $$ \sum (-1)^p[\wedge^pT^*M] \in K(M). ...
5
votes
0answers
217 views

Roots of unity in algebraic K-theory

For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit. For ...
3
votes
0answers
81 views

Torsion of Bass, Farrell and Waldhausen nil groups

Let H be an infinite virtually cyclic group. If H is orientable (resp. non-orientable) the Farrell nil groups $N_n(\mathbb{Z}H,\alpha) $ (resp. the Waldhausen nil groups ...
10
votes
1answer
480 views

K-Theory space of finite abelian groups

Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a ...
4
votes
4answers
509 views

Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles

I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that: i) they vanish if the bundle of unit ...
2
votes
0answers
201 views

k-theory of $\mathbb{Z}$

I have a doubt. Borel computed the rank of the higher algebraic k-theory of $\mathbb{Z}$: $rank(K_n)(\mathbb{Z})= 1$ if $n\equiv1 mod4$, otherwise this rank is equal to 0. On the other hand Bjorn ...
6
votes
2answers
285 views

Trivial cobordism group in dimensions 1, 3, 7 related to H-space structures on the spheres in these dimensions?

Is there a connection between the existence of H-space structures on $S^1$, $S^3$ and $S^7$ and the fact that every (closed) 1-manifold, 3-manifold and oriented 7-manifold is a boundary, or is this a ...
13
votes
2answers
900 views

Can anyone explain to me what is an assembly map?

Or can you give me a good place to read about things related to assembly map, besides wikipedia? I am specially interested in the case of algebraic K-theory. Would appreciated if you could provide ...
10
votes
1answer
686 views

Friedlander Conjecture

I just read on the ALGTOP discussion list that Morel has announced a proof of the Friedlander conjecture. Question: Are there other applications besides the Milnor conjecture $H_*(G,F_p)=H_*(BG,F_p)$ ...
13
votes
2answers
689 views

Brauer Groups and K-Theory

Is there some a priori reason why we should expect the Brauer group of real [complex] super vector spaces to be closely related to periodicity in real [complex] K-theory? By "a priori" I mean a proof ...
9
votes
1answer
402 views

Homology of infinite loop spaces $QX$

Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}_p$ is a Hopf algebra over the ...
0
votes
0answers
173 views

Topological K-theory of Bohr compactification of real numbers

I am interested in the K-theory of the Bohr compactification $\mathbb{R}_B$ of the real numbers. Do we have $K_0(C(\mathbb{R}_B))$ isomorphic to $K_1(C(\mathbb{R}_B))$ ? More generally, what do we ...
10
votes
2answers
994 views

Atiyah-Patodi-Singer Eta invariant and Chern-Simons form

I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant: 1) Is eta a topological invariant ...
0
votes
0answers
248 views

A modified version of K-theory for manifolds ?

If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
9
votes
0answers
377 views

Two constructions for BU×Z

Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$: 1) Take the groupoid of finite dimensional complex inner product spaces with isometries ...
8
votes
2answers
1k views

State of the art for Gersten's conjecture for K-theory?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to ...
16
votes
2answers
882 views

Algebraic K-theory of the group ring of the fundamental group

I know of two places where $K_{*}(\mathbb{Z}\pi_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology. The first is the Wall ...
2
votes
1answer
339 views

Spectral sequence for H-space bundles

Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle. One ...
11
votes
2answers
714 views

Twists of K-theory and tmf

I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How ...
10
votes
3answers
910 views

Is there a good definition of (topological) K-Theory over arbitrary spaces?

Hi (this is my very first question here, so please don't hurt me...) for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...
2
votes
1answer
179 views

pairs of matrices up to similiarity and vector bundles over punctured torus

I would like to construct 2D vector bundles over the punctured torus, but I don't know a lot of K-theory. Over the square, there can only be the trivial bundle, but now since ...
8
votes
0answers
239 views

H-space structure on the Calkin algebra

By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...