Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

**2**

votes

**1**answer

236 views

### What is the Euler characteristic of a mapping space?

Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...

**1**

vote

**0**answers

117 views

### the first chern class of complex vector bundles

Let $\xi^\mathbb{C}$ be a complex vector bundle over a manifold $M$ (or $CW$-complex $B$).
Case~1: $\xi^\mathbb{C}$ is a complex line bundle. Then the first Chern class
$c_1(\xi^\mathbb{C})$ is zero ...

**5**

votes

**2**answers

174 views

### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...

**7**

votes

**0**answers

110 views

### Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm.
If $X$ is compact,
Atiyah-Jänich proved that
...

**4**

votes

**0**answers

77 views

### Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...

**3**

votes

**1**answer

142 views

### Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this.
I wonder if there is any source telling about a ...

**1**

vote

**0**answers

53 views

### What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism ...

**9**

votes

**1**answer

266 views

### positions of a methane molecule with carbon atom at the origin

Let $\text{CH}_4$ be the molecule of Methane:
The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron.
Here we regard all atoms ...

**4**

votes

**1**answer

308 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays the sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classse as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...

**10**

votes

**2**answers

504 views

### Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes?

Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?
That is, for each spectrum $E$, do we have a lift in the ...

**9**

votes

**0**answers

122 views

### Mapping class groups in high dimension

Let $M$ be a $1$-connected, closed, smooth manifold with $dim(M)>4$ and let us set $MCG(M)=\pi_0(Diff(M))$. Dennis Sullivan proved that $MCG(M)$ is commensurable to an arithmetic group.
I was ...

**-4**

votes

**0**answers

59 views

### how to calculate $H_2(S^{1}\times S^{1}\times S^{1}\backslash S^{1}\times D)$? [on hold]

$S^{1}\times D$ is solid torus. $S^{1}\times D$ is submerged in $S^{1}\times S^{1}\times S^{1}$
Generally,
how to calculate $H_{2}(M\backslash S)$ or $b_{2}(M\backslash S)$ , where $b_{2}$ is ...

**10**

votes

**1**answer

218 views

### Reference request: sheaves on the site of d-manifolds

I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...

**9**

votes

**0**answers

96 views

### How does the HHR Norm functor interact with the cotensor over $G$-spaces?

Let $N_H^G$ be the norm functor from orthogonal $H$-spectra to orthogonal $G$-spectra. We know the category of orthogonal $G$-spectra $\mathcal{S}_G$ is enriched over the category of based $G$-spaces ...

**-2**

votes

**0**answers

102 views

### Homology of non-orientablre manifold [closed]

Is there any example of non-orientable n-manifold with boundary, we call this manifold M, with the property that $H_i(M, \mathbb Z_2)$ vanishes for all $0<i<n$? If this situation is impossible, ...

**1**

vote

**0**answers

126 views

### topology of $PSL(2, \mathbb C)$? [migrated]

What can one say about the topology of $\text{PSL}(2, \mathbb{C})$, for example its cohomology groups?
By using the long exact sequence in homotopy one can show that the inclusion $$\text{SO}_3 \to ...

**7**

votes

**3**answers

726 views

### Is there a notion of “flat vector bundle over a topological space”?

I am reading this paper and at the top of page 5 the author makes reference to categories consisting of flat complex vector bundles over $X$ where $X$ is an arbitrary topological space. However, the ...

**0**

votes

**0**answers

114 views

### All homotopy groups of infinite real, complex projective space, can someone supply reference? [closed]

What are all of the homotopy groups of infinite real projective space and infinite complex projective space? I need this result, but I could not find a direct reference in Hatcher as to their ...

**4**

votes

**0**answers

164 views

### Is $D(n)$ a Thom spectrum?

Fix $p=2$ and let $D(n)$ be a spectrum which filters the Eilenberg-Moore spectrum $H\mathbb{Z}/2$, i.e. $\mathrm{colim}\ D(n)=H\mathbb{Z}/2$. This spectrum can be considered as the cofibre of ...

**-4**

votes

**0**answers

146 views

### Almost complex structures on compact surfaces

Let $M$ be a real manifold of dimension $n$ and $E\rightarrow M$ be a rank two vector bundle above it. Let $J_0$ and $J_1$ be two almost complex structures on $E.$ If there is a continuous ...

**3**

votes

**2**answers

324 views

### Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...

**11**

votes

**2**answers

273 views

### Gerstenhaber conjecture for free loop space

I- Is the following statement still a conjecture see this article ?
Conjecture (?)
Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild ...

**8**

votes

**0**answers

106 views

### Inducing up the group homomorphism between mapping class groups

There are many ways to embed the braid group into the mapping class group of a surface. To describe one of them, let ${C}_{2g+2}(\mathbb{D}^2)$ be the configuration of unordered $2g+2$ points in the ...

**3**

votes

**1**answer

117 views

### very basic question on p-typical group law and MU and BP

I am a beginner trying to understand Dan Quillen's idempotent map $\xi:MU_{(p)} \to MU_{(p)}$ described in his classic paper 5-page paper On formal group laws...
Let $F$ be a formal group law over a ...

**3**

votes

**0**answers

131 views

### What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis
for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion).
Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...

**7**

votes

**0**answers

152 views

### Is there a definition of an $\infty$-groupoid in HoTT whose terms are $n$-manifolds and whose higher morphisms are diffeomorphisms/isotopies/etc?

Suppose you want to work with TQFTs in homotopy type theory (HoTT). Working with $(\infty,n)$-categories, or even $(\infty,1)$-categories, is something that I gather is too difficult for HoTT at the ...

**6**

votes

**0**answers

172 views

### Mapping class group action on fundamental group of punctured elliptic curves

Let $(\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}}$ be the moduli stack of elliptic curves over $\overline{\mathbb{Q}}$. By Oda, we know that its etale fundamental group is $\widehat{SL_2(\mathbb{Z})}$.
...

**4**

votes

**2**answers

327 views

### Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures?
Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...

**9**

votes

**0**answers

135 views

### Stiefel-Whitney class of fibre bundles

Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...

**7**

votes

**0**answers

66 views

### configuration space of Riemannian manifolds with a parameter on the distance of distinct points

Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, ...

**5**

votes

**1**answer

169 views

### How can we explicitly verify a canonical Dirichlet domain for this hyperbolic punctured torus

This is a follow-up question to a question from math.stackexchange: http://math.stackexchange.com/q/1436253/67563
Had it not been for the exchange there between myself and @Lee_Mosher in the comments ...

**8**

votes

**0**answers

149 views

### Geometric argument for “easy” part of Jordan-Brouwer separation theorem without local flatness

Let $M^n \subset \mathbb{R}^{n+1}$ be an $n$-dimensional compact connected topological submanifold. The Jordan-Brouwer separation theorem says that $\mathbb{R}^{n+1} \setminus M^n$ contains two ...

**2**

votes

**1**answer

129 views

### Universal covering and double cover functors

Initially posted on MSE
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...

**12**

votes

**2**answers

371 views

### Non-realizability of $\mathbb{Q}$ as a cohomology group

(This is a re-post of 1)
In the paper "On the realizability of singular cohomology groups" by Kan and Whitehead, it is shown that there is no space $X$ and integer $n\geq 1$ such that $H^{n−1}(X)=0$ ...

**5**

votes

**2**answers

183 views

### Are there non-contractible spaces $A$ and $B$ such that $A \wedge B$ is contractible?

The question is in the title : Can we find spaces $A$ and $B$, each non contractible, such that their smash product $A \wedge B$, i.e. the homotopy cofibre of $A \vee B \to A \times B$, is a ...

**6**

votes

**1**answer

182 views

### Can the standard map $\Sigma \Omega X \to X$ be a homotopy equivalence?

The question is in the title : are there spaces X such that the adjoint of the identity on the loop space $\Omega X$, i.e. $\Sigma\Omega X \to X$, is a homotopy equivalence ?

**5**

votes

**1**answer

313 views

### Iterated Homotopy Quotient

If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can ...

**6**

votes

**2**answers

259 views

### Seifert--van Kampen for the loop space dga

I recently noticed that I could mostly prove a special case of the following statement. I think it's true in general, though perhaps only for nice spaces.
Let $X$ be a topological space, and ...

**6**

votes

**1**answer

202 views

### canonical action of symmetric groups on orthogonal groups

There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$:
$$
S_{n+1}\to O(n)
$$
given as follows.
(1). I regard $O(n)$ as the isometry group of the unit ...

**6**

votes

**3**answers

296 views

### Examples of Stiefel-Whitney classes of manifolds

Let $M$ by an compact, connected $n$-dimensional manifold without boundary.
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, ...

**5**

votes

**1**answer

107 views

### symmetric group of regular polyhedrons

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let ...

**12**

votes

**2**answers

463 views

### formula for Eta invariant

Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...

**1**

vote

**1**answer

191 views

### Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the Borsuk Ulam theorem
for the Euclidean n-sphere that is based only on the theory
of Banach algebras. I checked on MR but had no success.

**1**

vote

**0**answers

102 views

### Cofibre of the $n$-fold transfer $\mathbb{R}P_+^{\wedge n}\to S^0$

I want to know what is known about the cofibre of the $n$-fold transfer map $\mathbb{R}P^{\wedge n}_+\to S^0$, for $n>1$. I am happy to know of any specific example worked out. The case $n=1$ is ...

**6**

votes

**0**answers

92 views

### cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...

**16**

votes

**5**answers

805 views

### What are examples when the equality of some invariants is good enough in algebraic topology?

As far as my understanding goes, most of the tools of algebraic topology (homotopy groups, homology groups, cup product, cohomology operations, Hopf invariant, signature, characteristic classes, knot ...

**10**

votes

**2**answers

379 views

### Do there exist “topologically significant” (and not “algebraic”) triangulated categories killed by the multiplication by $p$?

I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably ...

**19**

votes

**0**answers

382 views

### Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...

**7**

votes

**1**answer

281 views

### electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...

**3**

votes

**2**answers

263 views

### Is it possible to compute coefficients of the formal group of an elliptic curve?

This may be trivial but I cannot find it. Given an elliptic curve, $C$ over $R$ with chosen parametrization $R\to \mathbb{A}^5_{\mathbb{Z}}=Spec(A)$, is there a way to compute coefficients of the ...