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

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        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_{\...
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        1answer
        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 ...
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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 \...
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        0answers
        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 ...
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        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 ...
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        1answer
        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}...
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        0answers
        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 ...
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        1answer
        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 ...
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        votes
        1answer
        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{...
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        1answer
        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 ...
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        vote
        1answer
        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 ...
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        0answers
        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. ...
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        2answers
        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 ...
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        0answers
        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 ...
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        votes
        1answer
        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 ...

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        山西福彩快乐十分钟
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