All Questions
Tagged with dg.differential-geometry kt.k-theory-and-homology
20 questions with no upvoted or accepted answers
50
votes
0
answers
12k
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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
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
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
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
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
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
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
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