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

**2**

votes

**2**answers

196 views

### Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?

In the web page http://www.encyclopediaofmath.org/index.php/Moore_space it can be found the following statement:
If $K(\mathbb Z,n)$ is the Eilenberg–MacLane space of the group of integers ...

**4**

votes

**0**answers

50 views

### Multi-podal points

Two points $x,y \in \mathbb{R}^n$ are called antipodal if $x = -y$.
Stated differently, $x,y$ are antipodal if:
They have the same absolute value in each of their $n$ coordinates;
Each of their ...

**23**

votes

**1**answer

703 views

### When is there a submersion from a sphere into a sphere?

(First posted on math.SE, with no answers.)
That is:
For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$?
The discussion at this math.SE question has ...

**2**

votes

**0**answers

73 views

### Naive G-spectrum representing geometric equivariant cobordism

Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum.
...

**2**

votes

**2**answers

166 views

### Convergence of a sum with the ranks of homotopy groups

Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. ...

**0**

votes

**0**answers

45 views

### cobordism and smoth-manifold [on hold]

Let M, N , N' and M' be smooth n-manifolds with nonempty boundaries , and suppose h:∂M→∂N , g:∂M'→∂N' are diffeomorphisms . Let M∪_h N be the adjunction space formed by identifying each xϵ∂M with ...

**11**

votes

**6**answers

2k views

### Homology of loop space

I've been reading Galatius's Park City notes on the Madsen-Weiss theorem (available here).
On page 8, he states the following theorem. Let $X$ be a space such that $\pi_1(X)$ is abelian and acts ...

**65**

votes

**17**answers

9k views

### Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...

**9**

votes

**1**answer

198 views

### Alexander duality for non-manifolds

Let $X$ be a CW complex and $A$ a subcomplex. I will assume that both are compact, and that $X$ is $n$-dimensional. Furthermore, assume that the local homology of $X$ is that of a manifold in a ...

**8**

votes

**1**answer

203 views

### Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$?

Does there exist a smooth, closed, non-orientable $6$-manifold $M$ such that $H_4(M;\mathbb{Z})=\mathbb{Z}/2$?

**5**

votes

**1**answer

223 views

### Simply connected CW-complex with only finitely many nontrivial homotopy and homology groups

Let $X$ be a simply connected CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily contractible?
I ...

**0**

votes

**0**answers

67 views

### a construction on Stiefel manifolds

Are there any references concerning the following space $V(k,N,X)$ and $U(k,N,X)$? And the cohomology of these spaces? Thanks.

**3**

votes

**1**answer

128 views

### Which automorphisms on $H_{1}(M^{3})$ are induced by homotopy equivalences?

Let $M^{3}$ be a closed orientable 3-manifold, and $\phi:H_{1}(M;\mathbb{Z})\to H_{1}(M;\mathbb{Z})$ be an automorphism of abelian groups.
My question is: Is there any characterization of $\phi$ ...

**2**

votes

**0**answers

84 views

### The most general set-up for tensors and connections

This is maybe a too vague question, so I will try to be as specific as possible. My question is:
What is the most general set-up where one can define tensors and connections?
For example, we know ...

**17**

votes

**3**answers

686 views

### Degrees of self-maps of aspherical manifolds

In "Infinitesimal computations in Topology", Publ IHES, page 318, Dennis Sullivan writes "Recall any self-mapping
of a Riemann surface of genus $g>1$ either has
degree $0$ or degree $\pm 1$." ...

**11**

votes

**1**answer

163 views

### The multiplication on $THH$ of finite fields

Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...

**3**

votes

**1**answer

165 views

### Generalization of Borsuk-Ulam to arbitrary ratio

Let $g: S^n \to R^n$ be a continuous odd function (i.e. $g(-x)=-g(x)$ for all $x$). The Borsuk-Ulam theorem implies that $g$ has a zero, i.e. there is an $x$ such that $g(x)=(0,0,...,0)$.
Suppose $g$ ...

**10**

votes

**1**answer

140 views

### K-groups of a permutative category - are they finite?

Let $\mathcal C$ be a permutative category, that is a symmetrical monoidal category with strict associativity. One can then define the $K$-groups of $\mathcal C$, for $n >0$ by
$$K_n(\mathcal C) = ...

**15**

votes

**1**answer

494 views

### Is every degree 1 self-map a homotopy equivalence?

In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this ...

**5**

votes

**2**answers

207 views

### The Borel construction of equivariant cobordism

In $K$ theory, the Borel construction of equivariant cohomology is somehow not the right one. The $G$-equivaraint $K$ theory of a point should be the representation ring of $G$, but $K(BG)$ is this ...

**3**

votes

**0**answers

96 views

### Inverse limit in shape theory

Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms ...

**7**

votes

**0**answers

219 views

### Why does $Mf$ always support an $Mf$-orientation?

Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom ...

**6**

votes

**4**answers

612 views

### Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?

That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...

**1**

vote

**1**answer

190 views

### What is the cokernel of the map $H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow H^2(X,\mathbb Z).$

For a manifold $X$ (for simplicity, assumed to be compact), let $\pi_1(X)$ be the fundamental group of $X$. What is the cokernel of the map $$H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow ...

**18**

votes

**1**answer

886 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

**8**

votes

**2**answers

421 views

### Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$

Let $X$ be a nice topological space and denote by $\pi_1(X)$ its fundamental group.
It is well-known that there is a well-defined map
$$
0 \rightarrow H^2(\pi_1(X),A) \rightarrow H^2(X,A),$$
where ...

**14**

votes

**3**answers

527 views

### When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$

For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case ...

**5**

votes

**1**answer

212 views

### When are configuration spaces aspherical?

It is a theorem of Fox and Neuwirth that the space $C_k \mathbb R^2$ of unordered configurations of $k$ points in $\mathbb R^2$ is apsherical, i.e. has trivial higher homotopy groups.
This has some ...

**7**

votes

**0**answers

160 views

### Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...

**3**

votes

**1**answer

138 views

### Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems
'':
Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every
$n\in \omega, ...

**3**

votes

**1**answer

83 views

### If $\cup (A_i \cup -A_i)=\mathbb S^d$, then is there an $x$ in $d$ sets?

If for some collection of open sets $\cup_{i\in I} (A_i \cup^* -A_i)=\mathbb S^d$, then is there an $x\in \mathbb S^d$ and $i_1,\ldots i_d\in I$ for which $x\in A_{i_1}\cap \ldots \cap A_{i_d}$?
...

**0**

votes

**0**answers

125 views

### Hochschild-Serre spectral sequence

The Hochschild-Serre spectral sequence says that for a short exact sequence $$1 \to G \to H \to K \to 1 \quad (1)$$ of (discrete) groups, we have a first quadrant spectral sequence with $E_2$ page
...

**4**

votes

**2**answers

261 views

### The classifying space of an infinite totally ordered set is contractible

I asked this question on math.stackexchange, but no one answered.
Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...

**7**

votes

**0**answers

173 views

### May's infinite loop machine for Friedlander's result for Adams conjecture

Eric M. Friedlander in the paper The infinite loop Adams conjecture via Classification Theorem for $\Gamma$-spaces proved the infinite loop Adams conjecture using techniques involved $\Gamma$-space.
...

**4**

votes

**0**answers

229 views

### Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces

Warmup:
Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric ...

**0**

votes

**0**answers

40 views

### Looking for Information on Local Degree of Maps on Homology Manifolds

By a homology $n$-manifold, we mean a topological space $X$ such that for all $x \in X$:
1: if $k \neq n$ then $H_k(X, X-x)=0$
2: $H_n(X,X-x) \cong \mathbb{Z}$.
Given homology $n$-manifolds $X$ and ...

**12**

votes

**1**answer

415 views

### Condition on a Hopf operad for tensor product in the base category to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**5**

votes

**1**answer

459 views

### How do small changes in a filtered complex affect the associated spectral sequence

I have recently been learning about spectral sequences, not in the context of any problem, but mostly out of curiosity. There are two questions that have occurred to me which I've been unable to ...

**3**

votes

**0**answers

82 views

### Maps between equivariant loop spaces

I have an elementary question about equivariant loop spaces that I feel it should be well known.
Given a finite group $G$ and a finite $G$-set $J$ let $S^J=\mathbb{R}[J]^+$ be the permutation ...

**3**

votes

**1**answer

77 views

### Loop space of Fredholm operators from a Relative loop space

Atiyah and Singer proved that the nontrivial component of the set of skew-adjoint Fredholm operators $ \hat{\mathcal{F}_{*}}(\mathscr{H})$ is homotopic to the loop space of Fredholm operators ...

**4**

votes

**0**answers

129 views

### Fundamental groups of stably parallelizable manifolds

Is it possible to realize every finitely presented solvable group as a fundamental group of a stably parallelizable closed n-manifold? If not, are there any known counterexamples?

**1**

vote

**1**answer

150 views

### Homology of manifold with action of group

Sorry for my ignorance in advance, this should be a very naive question and I would be happy for a reference.
Let $G$ be an arbitrary group (not necessary finite) acting on two (connected) manifolds ...

**5**

votes

**1**answer

186 views

### When does a map in the stable homotopy group gets killed when smashed with cone of itself?

Consider an element $e \in \pi_n^s(S^0)$, in the stable homotopy groups of sphere. Let $C$ denote the spectrum which is cone of $e$, i.e. $C$ fits in the cofiber sequence
$$ S^n \to S^0 \to C.$$
...

**7**

votes

**2**answers

300 views

### Correspondence between operads and monads requires tensor distribute over coproduct?

In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...

**5**

votes

**1**answer

180 views

### Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold

Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing ...

**3**

votes

**1**answer

191 views

### Dehn-Sommerville relations for $\Delta$-complexes

Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the ...

**1**

vote

**0**answers

109 views

### Representations and K-theory of a finite group

This question is motivated by the calculation of the higher algebraic $K$-groups of a finite field.
Let $G$ be a finite group, the case I am most interested in is $G = \text{Gl}_n(\mathbb F_q)$, but ...

**-1**

votes

**1**answer

83 views

### unordered configuration space of pointed space

Let $(X,*)$ be a pointed topological space.
Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid x_i\neq x_j, i\neq j\}$.
Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.
Is there an inclusion ...

**7**

votes

**2**answers

517 views

### Differentials in the Adams Spectral Sequence for spheres at the prime p=2

How does one compute the differentials in the Adams Spectral Sequence for spheres at the prime 2 in the range $13\le t-s\le 20$? There seem to be 6 nonzero differentials, and at this point I only ...

**6**

votes

**3**answers

2k views

### How are fiber bundles, transition functions and principal bundles related?

Please read the edit below.
Is my understanding of this correct? Fix a sufficiently nice and connected topological space $B$ and a topological group $G$. A principal bundle $E\to B$ with structure ...