All Questions
Tagged with kt.k-theory-and-homology dg.differential-geometry
47 questions
62
votes
3
answers
6k
views
Atiyah-Singer theorem-a big picture
So far I made several attempts to really learn Atiyah-Singer theorem. In order
to really understand this result a rather broad background is required: you need
to know analysis (pseudodifferential ...
- 9,330
53
votes
12
answers
9k
views
Looking for an introduction to orbifolds
Is there any source where the basic facts about orbifolds are written and proved in full detail?
I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more ...
50
votes
0
answers
12k
views
Atiyah's paper on complex structures on $S^6$
M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
https://arxiv.org/abs/1610.09366
It relies on the topological $K$-theory $KR$ and in ...
- 9,870
48
votes
0
answers
17k
views
What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
- 2,995
33
votes
2
answers
2k
views
What are the "correct" conventions for defining Clifford algebras?
I have three related questions about conventions for defining Clifford algebras.
1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...
- 118k
21
votes
2
answers
2k
views
Applications of Atiyah-Singer using pseudodifferential operators
Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...
- 2,998
18
votes
3
answers
3k
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 (...
- 405
18
votes
3
answers
2k
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 the paper "Exotic ...
- 875
11
votes
1
answer
2k
views
A survey for various $K$-homology theories and their relationship
The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory....
- 593
10
votes
0
answers
6k
views
Atiyah's paper "Non-existent complex 6-sphere"
I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions.
Consider the ...
- 171
9
votes
3
answers
1k
views
Integration in equivariant K-theory
Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the ...
- 226
9
votes
1
answer
756
views
Does there exist a GRR-like generalization of the AS Index Theorem?
The Hirzebruch Riemann-Roch Theorem (HRR) expresses an analytic/algebraic invariant, namely the Euler-Poincaré characteristic of a vector bundle $V$ over a compact complex/algebraic manifold $X$, as ...
- 1,020
8
votes
1
answer
318
views
K-homology classes of Dirac operators on Hermitian manifolds
Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely
1) (d + d$^*,\Omega^{*})$
2) ($\partial$ + $\...
- 1,453
7
votes
1
answer
485
views
Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2
Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?
- 1,236
7
votes
0
answers
270
views
The Todd class and Weyl's character formula
Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...
- 1,379
7
votes
0
answers
359
views
Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices
Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...
- 186
6
votes
2
answers
279
views
Differential structures and K-homology groups
What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...
- 167
6
votes
1
answer
465
views
Splitting principle in equivariant cohomology
The following is a weaker version of what is called splitting principle in
Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6:
Let $G$ be a compact ...
- 103
6
votes
0
answers
167
views
Elliptic operators with with same index but non homotopic symbols
Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$.
Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold.
In Atiyah-Singer "the index of ...
- 2,533
6
votes
0
answers
170
views
Does the $K^1$-group of a complete flag variety vanish?
For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space
$$
U(n)/T^n
$$
is called the complete flag variety of order $n$. For the special ...
- 141
6
votes
0
answers
230
views
Equivariant Venice Lemma
In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as
Theorem: For ...
- 386
5
votes
1
answer
543
views
The principal symbol as an element in the K-theory
This line
The symbol may naturally be thought of as an element in the K-theory
of X
appears in the nLab page on principal symbols for differential operators. What does this mean? Are they talking ...
- 1,144
5
votes
1
answer
772
views
Does bundle with torsion Chern classes admit flat connection?
I want to know something about torsion in topological k-theory. So, consider complex bundle with chern classes lying in torsion part of integer homologies and my question is : does it admit a flat ...
- 789
5
votes
1
answer
308
views
Compactly supported chern character
It is a standard result that for a CW complex $X$, the chern character
$$\text{ch}: K^*(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^*(X,\mathbb{Q})$$
induces an isomorphism. Suppose now that $X$ is an open ...
- 988
5
votes
1
answer
392
views
Equivariant $K$-theory, singular vectors, and flag manifolds
For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...
- 3,399
5
votes
1
answer
366
views
K-theory for a (geometric) stack
There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...
- 6,023
5
votes
0
answers
297
views
Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
- 2,247
5
votes
0
answers
238
views
Tensor product of "difference bundles" ( Atiyah construction)
There is a well-known in index theory "difference bundle" construction of Atiyah( see for example the original paper "Clifford modules"). And also there is a corresponding formula for the tensor ...
- 51
4
votes
2
answers
633
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 $K_{(...
- 1,173
4
votes
1
answer
384
views
Torsion In $K$ theory on simply connected manifolds
The usual construction for finding torsion elements on complex $K$ theory is using flat vector bundles. So is it still possible to find a simply connected compact space with a nonzero torsion in its $...
- 435
4
votes
1
answer
236
views
Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma
I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...
- 9,330
4
votes
1
answer
827
views
second fundamental form
Hi all,
Currently I'm reading a paper about the geometry of Grassmannians:
www.omup.jp/modules/papers/riemann/04Nagatomo.pdf
In there, the author regards the second fundamental form of the k-...
4
votes
0
answers
107
views
KK-theory for commutative $C^*$-algebras
The Gelfand--Naimark theorem tells us to regard noncommutative $C^*$-algebras as "noncommutative function spaces". In that spirit $K$-theory the Grothendieck group of "noncommutative ...
- 1,144
4
votes
0
answers
148
views
Definition of the $G$-equivariant index map
My question concerns a statement on page 12 of the following paper of Baum, Connes, and Higson:
http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf
about the definition ...
- 1,903
3
votes
1
answer
544
views
Chern classes of generators of $K(S^{2n})$
Calculate the Chern classes $$ c_n \in H^{2n}(S^{2n})$$ for the generator of the group $$ K(S^{2n})$$ where $S^{2n} $ - sphere of dimension $ 2n $, $ K(S^{2n})$ - group from K-theory.
I found the ...
- 31
3
votes
1
answer
319
views
Slice theorem for proper groupoids
Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$.
Assume $G$ is étale, i.e., the source and range maps of $G$ are local ...
- 410
2
votes
2
answers
124
views
Invertible (isometric) sections of certain hom bundles over sphere
Assume that we have a vector bundle $E$ over $S^n$.
Is there a continuous family of invertible linear maps $T_x:E_x \to E_{-x}$?
Here continuity has the obvious meaning as soon as ...
- 356
2
votes
1
answer
246
views
$C^*$-algebras appearance in study of Lie groupoids and differentiable stacks
I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...
- 6,023
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
2
votes
0
answers
286
views
Open problems in the theory of manifolds relating to construction [closed]
A while ago I stumbled across a paper of Thurston: Some Simple Examples of Symplectic Manifolds, where Thurston constructs closed symplectic manifolds with no Kaehler structure. My question is: What ...
2
votes
0
answers
105
views
Multiplicativity of the analytic index (or of kernel bundle)
What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
- 1,173
1
vote
0
answers
240
views
Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
- 356
1
vote
0
answers
172
views
Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563
1
vote
0
answers
132
views
A possible kind of $K$ theory via comparison of sphere bundles associated to given vector bundles
Let $E$ be a vector bundle on a topological space $X$.Thanks to Allen Hatcher's book "Vector Bundles and K theory", the construction of sphere bundle $S(E)$ can be done without any inner ...
- 356
1
vote
0
answers
965
views
Trivial normal bundle
I would like to know if there is a theorem along those lines: let $V$ be a submanifold in $\mathbb{R}^n$ such that $V$ is the boundary of a submanifold with boundary $W$. Then, the normal bundle of $V$...
- 11
1
vote
0
answers
81
views
Ring structure for $K^{-1}$?
My questions are
whether there exists a product structure for $K^{-1}(X)$? Here $K^{-1}$ is the odd topological $K$-group, and $X$ is a compact space (or a manifold), say.
If such a ring structure ...
- 1,173
0
votes
0
answers
307
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 ...
- 505