All Questions
9,056 questions
23
votes
2
answers
881
views
Vanishing of characteristic numbers vs vanishing of characteristic classes
A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...
- 521
23
votes
2
answers
850
views
Classification of fake (quaternionic, octonionic) projective spaces
If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...
- 9,580
23
votes
2
answers
3k
views
Calculating Mayer-Vietoris efficiently
This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere....
- 156k
23
votes
1
answer
718
views
Del Pezzo surfaces and homotopy groups of spheres
A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...
- 6,920
23
votes
1
answer
4k
views
The Dedekind eta function in physics
This interesting little fellow (a nice introduction is the video "Mock Modular Forms are Everywhere" by Cheng and Felder) popped up in some operator algebra (Witt / Virasoro Lie algebra) I ...
23
votes
0
answers
717
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essentially different"?
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize ...
- 118k
23
votes
0
answers
463
views
Topological loops vs. algebro-geometric suspension in Hochschild homology
Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...
- 6,069
23
votes
0
answers
590
views
What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra?
There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are ...
- 118k
23
votes
0
answers
784
views
Characteristic classes for $E_8$ bundles
$\DeclareMathOperator\B{B}\DeclareMathOperator\SU{SU}$Given a principal $E_8$ bundle $P\rightarrow X$ one can take the
adjoint representation $\rho :E_8\rightarrow \SU(\mathbb C^{248})$
and form the ...
- 694
22
votes
4
answers
4k
views
Why study simplicial homotopy groups?
The standard definition for simplicial homotopy groups only works for Kan complexes (cf. http://ncatlab.org/nlab/show/simplicial+homotopy+group). I learned that the hard way, when I tried to compute a ...
- 1,558
22
votes
7
answers
3k
views
Essential theorems in group (co)homology
I'm trying to fill in the gaps in my understanding of group (co)homology and I'm wondering what are considered the "must know" theorems and concepts. I'm thinking of things along the lines of
Hopf's ...
22
votes
2
answers
6k
views
References and resources for (learning) chromatic homotopy theory and related areas
What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?
22
votes
5
answers
3k
views
Is $L^p(\mathbb{R})$ minus the zero function contractible?
Is $L^p(\mathbb{R}) \setminus 0$ contractible? My intuition says that the answer is yes, but I'm afraid that this is based on thinking of this as somehow similar to a limit of $\mathbb{R}^n \setminus ...
- 433
22
votes
5
answers
5k
views
Is there a good way to understand the free loop space of a sphere?
I'd like to understand the structure of the free loop space of $S^n$ for small values of $n$. Here "understand" means roughly that I'd like to know a CW complex with the same homotopy type.
I ...
- 8,493
22
votes
5
answers
2k
views
Homeomorphisms of $S^n\times S^1$
Is every homeomorphism $h$ of $S^n\times S^1$ with identity action in $\pi_1$ pseudo isotopic to a homeomorphism $g$ of $S^n\times S^1$ such that $g(S^n\times x)=S^n\times x$ for each $x\in S^1$? I ...
- 339
22
votes
4
answers
3k
views
Betti numbers of moduli spaces of smooth Riemann surfaces
Where can I find a list of the known Betti numbers of the moduli spaces $\mathcal{M}_{g,n}$ of genus $g$ Riemann surfaces with $n$ marked points? I need it to cross check results by an implemented ...
- 6,777
22
votes
6
answers
3k
views
Does every vector bundle allow a finite trivialization cover?
Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...
- 1,284
22
votes
4
answers
3k
views
Why is the classifying space of the natural numbers homotopy equivalent to the circle?
Is there a direct way to seeing that $B{\mathbb{N}}\simeq S^1$, i.e. the classifying space of the monoid of natural numbers is homotopy equivalent to the circle?
Here, since the natural numbers ${\...
- 681
22
votes
4
answers
2k
views
Functorial Whitehead Tower?
The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice ...
- 27.5k
22
votes
4
answers
2k
views
fixed point property for maps of compacts
Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point.
Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...
- 31.2k
22
votes
6
answers
2k
views
Is any interesting question about a group G decidable from a presentation of G?
We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton. Thus F0 ...
- 1,219
22
votes
2
answers
978
views
Fixed-point free diffeomorphisms of surfaces fixing no homology classes
One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
- 313
22
votes
2
answers
977
views
Mapping from a finite index subgroup onto the whole group
Dear All,
here is the question:
Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$?
My guess ...
- 1,437
22
votes
2
answers
6k
views
Grothendieck's Tohoku Paper and Combinatorial Topology
I've read some discussions of Grothendieck's famous Tohoku Paper, and I understand that one reason it was a landmark paper was that it introduced abelian categories and gave us sheaf cohomology as a ...
- 221
22
votes
8
answers
2k
views
Examples of Brown (co)fibration categories that are not Quillen model categories?
K.S. Brown has shown that much of abstract homotopy theory can be carried out in the setting of Brown (co)fibration categories [MR0341469]. The decisive property, immediate from the axioms, is that ...
22
votes
5
answers
7k
views
Describing the universal covering map for the twice punctured complex plane
As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map.
In a sense, this shows that the logarithm has ...
- 5,530
22
votes
2
answers
1k
views
The image of the point-pushing group in the hyperelliptic representation of the braid group
Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation
$\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$
called the "hyperelliptic representation," which ...
- 19.2k
22
votes
3
answers
820
views
Boardman's thesis or mimeographed notes
I would like to know if there is some online source where Boardman's 1964 thesis is available or his Warwick mimeographed notes. This is because by what I've heard Boardman's construction has a more ...
- 12.1k
22
votes
1
answer
1k
views
Little disks operad and $Gal (\bar {Q}/Q)$
My question is simple:
How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?
I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling with ...
- 2,734
22
votes
3
answers
1k
views
Applications of topological and diferentiable stacks
What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well ...
- 15.5k
22
votes
2
answers
3k
views
Interpretation of elements of H^1 in sheaf cohomology.
Given a variety V and a locally free (coherent) sheaf $\mathcal{F}$ of rank 1 (equivalently a line bundle $L$), I can do a Cech cohomology on it. Then $H^0(V; \mathcal{F})$ are just global sections. ...
- 422
22
votes
2
answers
1k
views
What clues originally hinted at stability phenomena in algebraic topology?
If you didn't know anything about stabilization phenomena in algebraic topology and were trying to discover/prove theorems about the homotopy theory of spaces, what clues would point you toward ...
- 443
22
votes
2
answers
1k
views
Eversion of the 6-sphere in 7-space
Say that $S^n$ "admits eversion" if the inclusion $S^n \rightarrow \mathbb{R}^{n+1}$ is regularly homotopic to the antipodal map (where a "regular" homotopy is a continuous path through immersions).
...
- 732
22
votes
2
answers
1k
views
Toposes (topoi) as classifying toposes of groupoids
A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has ...
- 38.6k
22
votes
1
answer
679
views
When does rationalization commute with homotopy fixed points?
Let $X$ be a $G$-space. There are a number of places in the literature where one can find the claim that under certain conditions rationalization and taking homotopy fixed points with respect to a ...
- 8,167
22
votes
1
answer
805
views
Twistings for other cohomology theories
Twistings in cohomology theories have a long history and have been used to great effect. The classical example is cohomology with local coefficients. Using this one can formulate Poincaré duality and ...
- 8,167
22
votes
1
answer
719
views
What is the cohomological dimension of the commutator subgroup of the pure braid group?
I'm interested in computing the cohomological dimension of the commutator subgroup $[P_n,P_n]$ of the pure braid group $P_n$. I wasn't able to find a reference in the literature.
Because $[P_n,P_n]$ ...
22
votes
4
answers
2k
views
Natural setting for characteristic classes?
In my mind, algebraic topology is comprised of two components:
Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks".
...
- 22.1k
22
votes
1
answer
2k
views
The free loop space of spheres
Let $n>1$. The homology of the free loop space $\Lambda S^n$ of the sphere $S^n$ contains two torsion if $n$ is even. Thus the fibration
$$
\Omega S^n\rightarrow \Lambda S^n\rightarrow S^n
$$
is ...
- 7,583
22
votes
5
answers
4k
views
Why is complex projective space triangulable?
In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can ...
- 4,254
22
votes
2
answers
2k
views
Non standard Algebraic Topology
Let *$\mathbb R$ a field of non-standard real numbers (or any real closed field) equipped with its natural generalized metric $d(x,y)=|x-y|$. Equip *$\mathbb R^2$ and *$\mathbb R^3$ with the $\ell^1$-(...
- 6,034
22
votes
1
answer
1k
views
Word problem for fundamental group of submanifolds of the 4-sphere
Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$. For most constructions of these 4-...
- 44.4k
22
votes
1
answer
2k
views
Is algebraic $K$-theory a motivic spectrum?
I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy ...
- 64k
22
votes
1
answer
1k
views
What's with equivariant homotopy theory over a compact Lie group?
For some reason -- I'm not quite sure why -- I've developed the impression that I'm supposed to "tiptoe" around equivariant homotopy theory over groups that are not finite. Should I?
Let me explain. ...
- 64k
22
votes
1
answer
1k
views
Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
- 25.5k
22
votes
1
answer
2k
views
Formal-group interpretation for Lin's theorem?
Background
For compact Lie groups, Atiyah and Segal proved a strong relationship between Borel-equivariant K-theory, defined in terms of the K-theory of $X \times_G EG$, and the equivariant K-theory ...
- 52.6k
22
votes
2
answers
1k
views
Does $\mathrm{E}_7/(\mathrm{SU}_8/(\mathbb{Z}/2))$ carry an almost complex structure?
Recall the list of irreducible simply connected inner symmetric spaces of compact type in dimension $4k+2$:
Hermitian symmetric spaces (one can write them down explicitly);
Grassmannians of oriented ...
- 2,140
22
votes
2
answers
1k
views
Graphs, K-theory and combinatorial balls: conjectures
The following conjectures from Kapranov and Saito's Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions aren't as well-known as they aught to be, so I'd like to state ...
- 2,734
22
votes
1
answer
1k
views
How does Poincare duality interact with the Serre spectral sequence?
Suppose $F^m \to E^{m + n} \to B^n$ is a fiber bundle of closed oriented manifolds. I'm interested in understanding how the Serre spectral sequences for homology and cohomology of $E$ interact with ...
- 2,830
22
votes
1
answer
1k
views
Classical and Quantum Chern-Simons Theory
Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway.
Let $\Sigma$ be a two-manifold and $M$ a ...
- 13.6k