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        Questions tagged [cobordism]

        Cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold.

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        148 views
        +50

        On the algebraic cobordism cohomology of Eilenberg-MacLane spectrum

        Let $MU$ be the complex cobordism spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. In this question it is shown that the group $MU^{*}(H\mathbb{Z})$ is non trivial. From here, what can ...
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        184 views

        What mathematical background to i need in order to understand proofs of the h-cobordism theorem?

        I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some ...
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        votes
        1answer
        302 views

        Reference on complex cobordism

        I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism. ...
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        72 views

        Conformal group and cobordism

        In this post, I am exploring my thoughts on the implementation of conformal symmetry group structure and cobordism relations. Namely, I like to know what has been done and explored in the past? on ...
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        votes
        0answers
        64 views

        Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)

        My major question in this post here is that: How can we relate the following two mod 2 indices: $\eta$ invariant, the number of the zero modes of the Dirac operator $N_0'$ mod 2, associated to ...
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        148 views

        Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper

        In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
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        103 views

        Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

        This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post. I had discussed my computation of $$ \Omega_5^{...
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        votes
        1answer
        200 views

        Twisted spin bordism invariants in 5 dimensions

        [Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...
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        267 views

        How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?

        Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...
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        votes
        1answer
        384 views

        Critical dimensions D for “smooth manifolds iff triangulable manifolds”

        I am aware that at least for lower dimensions, "smooth manifolds iff triangulable manifolds" at least for dimensions below a certain critical dimensions D. My question is that for ...
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        votes
        1answer
        267 views

        Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?

        Let $k$ be a field of characteristic $0$. Let us denote by $\mathbf{1}_{k}$ the sphere spectrum. Let $MGL$ be the algebraic cobordism spectrum. We have the following diagram $$H\mathbb{Z}\...
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        69 views

        Decomposition of bordism groups for $BG$ where $G$ is a product of two groups

        Let a group $G=G_1 \times G_2$, where $G_1$ is a discrete group (can be finite or infinite), $G_2$ be any compact Lie group or finite group. Question: Is there some simple result that we can ...
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        132 views

        Categorification-like statement in the cobordism group?

        Suppose we consider a $d$-cobordism group classifying manifolds with $H$-structure and with a classifying space $BG$ of a group $G$, written as $$ \Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots, $$ ...
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        105 views

        Pin cobordism v.s. “KO” theory in low or in any dimensions

        Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion. This is related to a question and an answer supports the claim. Here we denote the $p$-...
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        149 views

        Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$

        We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of $$ \Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}), $$ where the $\mathbb{Z}/(8\mathbb{Z})$ is generated by a 2-manifold $M^...

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