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

**0**

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11 views

### Problem about the homology groups of the complex projective space $\mathbb{C}P_n$

My question is how we can compute the homology groups of the complex projective space $\mathbb{C}P_n$ by the following Corollary5.4 at page 31 in Milnor's book：
Corollary5.4
If $c_{\lambda+1}=c_{\...

**8**

votes

**1**answer

133 views

### Spectra with “finite” homology and homotopy

As known, any non-trivial finite spectrum $X$ can not have non-zero homotopy groups $\pi_i(X)$ only for finite number of $i$. As I understand, the same is true for any spectrum $X$ with finitely ...

**0**

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84 views

### CW-complex structure on the quotient

Let $X$ be an $n$-dimensional CW-complex and let $A \subseteq X$ be a subcomplex. I want to show that the quotient space $X/A$ admits a structure of a CW-complex with skeletons $(X/A)^j := \pi(X^j \...

**5**

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**0**answers

137 views

### Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space

In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background
"Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...

**8**

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**0**answers

82 views

### Relating bordism generators in d and d+2 dimensions — an explicit example

This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...

**2**

votes

**1**answer

96 views

### Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group

Let $M^d$ be a nontrivial bordism generator for the bordism group
$$
\Omega_d^G= \mathbb{Z}_n,
$$
suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...

**7**

votes

**0**answers

125 views

### “Gerbes” in the cobordism theory

In a lecture I attended today, I heard the use of gerbes in the cobordism theory.
Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group ...

**5**

votes

**1**answer

172 views

### Manifold generators of O-bordism invariants

If I understand correctly, I can obtain the $O$-cobordism group of
$$
\Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4,
$$
The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...

**3**

votes

**1**answer

238 views

### K-theory of finite diagram categories

Suppose $I$ is a finite $\infty$-category and $F:I\rightarrow\text{fCW}$ is a functor that takes values in finite CW complexes. For each $X\in I$, let $[F(X)]$ denote the class of $F(X)$ in $K_0(\text{...

**6**

votes

**1**answer

102 views

### Example of nonvanishing Waldhausen Nil group

In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen ...

**1**

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**1**answer

126 views

### What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure?

Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion ...

**1**

vote

**0**answers

109 views

### Example of open manifold with no free integer homology non-homeomorphic to a ball

I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.
...

**29**

votes

**2**answers

428 views

### Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...

**2**

votes

**0**answers

52 views

### Topological Shape Operator More Sensitive than Inverse Limits

This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...

**3**

votes

**1**answer

115 views

### Cellularity of anodyne extensions?

Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)
If not, is there a known counterexample?
Similarly, does ...