All Questions
9,056 questions
21
votes
3
answers
1k
views
What's the analogue of the Hilbert class field in the following analogy?
There's a wonderful analogy I've been trying to understand which asserts that field extensions are analogous to covering spaces, Galois groups are analogous to deck transformation groups, and ...
- 65.4k
5
votes
1
answer
383
views
Killing Chern classes
Let $G$ be a compact connected Lie group and let $E\to B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $G\to E$ ...
- 23.5k
8
votes
4
answers
586
views
Examples of the varying strengths of topological invariants
In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then ...
- 44.4k
15
votes
1
answer
1k
views
Stable ∞-categories as spectral categories
Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an ...
- 1,947
7
votes
2
answers
2k
views
categorical homotopy colimits
let $hTop_*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone ...
- 63.1k
3
votes
1
answer
858
views
Any reason why K_23(Z) has order 65520?
I'm rereading my notes and they mention that $K_{23}(\mathbb Z) = \mathbb Z/(65520)$
This looks like a good point to stop and ask whether there is any explanation for this $K$-group of integers (23 ...
- 803
8
votes
2
answers
431
views
Formulas for vector fields on Grassmannians?
The Wikipedia article on (real) Grassmannians gives a simple argument that the Euler characteristic satisfies a recurrence relation $$\chi G_{n,r} = \chi G_{n-1,r-1} + (-1)^r \chi G_{n-1,r}$$. This ...
CommunityBot
- 1
7
votes
4
answers
686
views
Realizing complexes with bases as cellular complexes
This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it.
Every free complex of abelian groups is isomorphic to the reduced cellular complex of some ...
- 4,736
7
votes
1
answer
282
views
Can you construct a mapping space from local data? (looking for reference)
I'd to know if/where there is a reference for the following construction.
Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
- 21k
3
votes
1
answer
299
views
disagreement between two definitions of the singular boundary map
Hi everyone, I have a little problem with the definition of singular boundary map in singular homology theory. It appears to be some disagreement between two authors. The first one is Hatcher in his '...
- 47.3k
11
votes
4
answers
958
views
Geometry of the multilagrangian Grassmannian
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
- 28.9k
18
votes
1
answer
943
views
Do chains and cochains know the same thing about the manifold?
This question was inspired by Poincaré quasi-isomorphism
Let $M$ be a closed oriented $n$-manifold. The cap product with the fundamental class of $M$ induces an isomorphism $H^i(M,\mathbf{Z})\to ...
CommunityBot
- 1
39
votes
3
answers
6k
views
Why do finite homotopy groups imply finite homology groups?
Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
- 20k
7
votes
2
answers
541
views
(Co-) Homology associated to Waldhausen K-Theory
Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...
- 52.6k
2
votes
2
answers
1k
views
Periodic mapping classes of the genus two orientable surface
Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and ...
- 28.2k
3
votes
3
answers
769
views
Reducible 3d torus bundles
Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,
could anyone give me a hint to classify them?
In contrast, do you agree ...
- 28.2k
2
votes
3
answers
746
views
Two solid N_3 glued by its boundary
Let $N_3$ be the genus three non orientable surface. Do we have an analogous 3d manifold as the solid torus and the solid Klein bottle for $N_3$? I don't see how to extend the ideas related to the 3d ...
- 345
21
votes
1
answer
767
views
The density hex
Gale famously showed that the determinacy of n-player, n-dimensional Hex is equivalent to the Brouwer fixed point theorem in n dimensions.
We can (and Gale does) view this as saying that if you d-...
- 24.7k
2
votes
1
answer
510
views
Are the C(S^n, S^n)'s homeomorphic ?
Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...
- 29.1k
16
votes
2
answers
817
views
Spin structures on 7-dimensional spherical space forms
Background
Let $M$ be a spin manifold and let $\Gamma$ be a finite group acting freely and isometrically on $M$ in such a way that $M/\Gamma$ is a smooth riemannian manifold. The quotient will be ...
- 33.3k
14
votes
1
answer
933
views
Smooth structures on PL 4-manifolds
Is it known whether $O(4) \to PL(4)$, the map from the orthogonal group to the group of piecewise linear homeomorphisms of $\mathbb{R}^4$, is a homotopy equivalence? By smoothing theory for PL ...
- 44.4k
7
votes
3
answers
1k
views
Joins of simplicial sets
Why doesn't the join operation on the category of simplicial sets commute up to unique isomorphism? I mean, aren't products and coproducts commutative up to isomorphism? That leads me to conclude at ...
- 21.3k
10
votes
1
answer
943
views
Cyclic spaces and S^1-equivariant homotopy theory
I'm trying to understand the relationship between cyclic spaces and S1-equivariant homotopy theory. More precisely, I only care about S1-spaces up to equivalence of fixed point spaces for the finite ...
- 25.2k
5
votes
2
answers
420
views
Connectivity after Geometric Realization?
Suppose that I have a map of simplicial spaces,
$ f: X_* \to Y_*$,
and that I know that the map on zero spaces $f_0: X_0 \to Y_0$ is n-connected. Can I conclude anything about the connectivity of ...
- 18.2k
1
vote
1
answer
256
views
N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms
It is well know that the genus three non orientable surface, N3, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N4 is the first non orientable surface with pseudo Anosov ...
- 345
15
votes
3
answers
2k
views
complex cobordism from formal group laws?
Reading Ravenel's "green book", I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. ...
- 781
3
votes
7
answers
4k
views
de Rham Cohomology of surfaces
Does anyone know a good book where I can find the computation of the de Rham Cohomology of surfaces in R^3 and other classical manifolds (higher dimensional spheres and projective spaces for example) ?...
- 496
37
votes
3
answers
5k
views
Topological Langlands?
In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...
CommunityBot
- 1
24
votes
1
answer
3k
views
Characteristic classes of sphere bundles over spheres in terms of clutching functions
I'm trying to understand Milnor's proof of the existence of exotic 7-spheres.
Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be ...
- 5,530
6
votes
4
answers
1k
views
What is known about the intersection pairing on H^{mid}?
When we restrict to the torsion-free part of the cohomology of a manifold, the intersection pairing is nondegenerate. In dimension 2n, this gives a bilinear form on the free part of Hn (symmetric if ...
- 6,247
0
votes
1
answer
314
views
Homology of symmetric groups
Let $S_n$ denote the symmetric group on $n$ letters, and let $S_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H_i(S_n(p))$ in $H_i(S_n)$ the $p$-primary part of $H_i(S_n)$?
- 47.3k
3
votes
1
answer
361
views
Is the coproduct of fibrant spectra fibrant again?
Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...
- 52.6k
1
vote
1
answer
870
views
Simplicial set notation and vocabulary question.
Notation question:
What does $(\Delta^1)^{ \{1, \ldots,n-1 \}}$ denote? UPDATE: I (David Speyer) tried to fix the LaTeX. Please see if I got it right.
Vocabulary question:
Suppose $z:\Delta^{n+1} \...
- 45.7k
3
votes
1
answer
928
views
Simple applications of Atiyah-Bott localization
I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.
Do you know any good ones?
- 156k
4
votes
3
answers
2k
views
free homotopy groups -- when do they exist?
Let (X,x) be a pointed space. There is an action of π1(X,x) on πn(X,x) -- determined by considering πn(X,x)=πn-1(ΩxX,x), where ΩxX denotes the space of loops in X based at x, ...
- 15.1k
8
votes
1
answer
637
views
Cohomology map induced by the group actions on homogeneous vector bundles
Here is a topological question which seems quite elementary. The answer to this question may be useful e.g. in estimating the orders of the automorphism groups of some algebraic varieties and in ...
- 52.6k
27
votes
3
answers
4k
views
"Dirty" proof that Eilenberg-MacLane spaces represent cohomology?
The standard approach to proving that $H^n(X; G)$ is represented by $K(G, n)$ seems to be to prove that $\text{Hom}(X, K(G, n))$ defines a cohomology theory and then use Eilenberg-Steenrod uniqueness. ...
- 141
7
votes
2
answers
268
views
What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)
Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...
- 51
28
votes
4
answers
4k
views
(∞, 1)-categorical description of equivariant homotopy theory
I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
- 3,685
6
votes
2
answers
396
views
Reference for iterated homotopy fixed points?
What are (good) references for results about iterated homotopy fixed points? That is, suppose G is a topological group acting on a space (or spectrum) X, and H is a normal subgroup of G. Then one ...
- 24.1k
12
votes
3
answers
4k
views
Notions of degree for maps $S^n \to S^n$?
In algebraic topology, we define a degree for a map $f: S^n \to S^n$ as where the induced map $f_*$ on the $n$-th homology group of $S^n$ sends $1$.
In differential topology, we have a different (...
- 156k
14
votes
2
answers
947
views
squares in stable homotopy
I noticed that the generator of the second stable stem b is the square of the generator of the first stable stem a, in the sense that if take two copies of a and smash product them together you get b ...
- 3,103
11
votes
2
answers
1k
views
Elliptic curve over spectra?
Filling the gaps in my knowledge to understand the tmf question.
So, what is the analogue of elliptic curve over the category of spectra?
CommunityBot
- 1
9
votes
1
answer
875
views
Analogue of Sperner's lemma for Lefschetz theorem?
Sorry if this is easy/well-known, I don't know much algebraic topology and I'm just curious about this question.
One of the easier proofs of the Brouwer fixed-point theorem (we'll say for n = 2 for ...
- 24.7k
5
votes
1
answer
190
views
Adapting families of diffeomorphisms to an open cover
Has anyone seen the following result in the literature? I've asked a few experts but so far I've come up with nothing.
Given a manifold M and an open cover {U_i} ...
CommunityBot
- 1
4
votes
1
answer
1k
views
properly interpreting Pi_0 in the homotopy exact sequence
Define the lens space L(m,n) as the quotient of S2m+1 by the action of the cyclic group ℤn⊂S1⊂ℂ*. We can create the infinite lens space L(∞,n) by a telescoping construction ...
- 27.2k
14
votes
2
answers
984
views
Recovering a monoidal category from its category of monoids
What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original ...
- 27.7k
13
votes
5
answers
1k
views
What kind of geometric operations "scale up" cohomology?
There's an obvious operation on the category of graded rings, given by "scaling up," multiplying the grading of every element by some fixed constant.
Does anyone know of an operation on the level of ...
- 26.8k
7
votes
2
answers
1k
views
One Point Compactification
Suppose X is a path-connected, locally compact, Hausdorff space and Y is its one-point compactification. Let G be the fundamental group of X and H be the fundamental group of Y. Is it true that the ...
- 25.2k
4
votes
2
answers
2k
views
(how) are vector bundles and homotopy groups related?
Hello,
homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the ...
- 31.2k