The goal of this thesis is to prove that π4(S) ' Z/2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πk(S) with k < n, and...

In a previous paper [5], the authors investigated extensively the Landweber-Araki Real cobordism spectrum MR ([1, 7]), and its stable summand at p = 2 called BPR. An Adams-like spectral sequence based on BPR was constructed, and homotopy groups of many spectra related to BPR were calculated. The goal of the present note is to begin investigating the question as to how BPR-theory contributes to ...

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+ 1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that for N = R the cobordism group of simple fold maps is isomorphic to the direct sum of the (n−1)th stable homotopy group of spheres and the (n − 1)th stable homotopy group o...

The quadric Q2n is the Z-scheme defined by equation ?i=1nxiyi=z(1?z). We show that a homogeneous space for split reductive group scheme SO2n+1 over Z. known to have A1-homotopy type of motivic sphere and identification as allows us give characteristic independent affine representability statement spheres. This last observation comparison results between Chow–Witt groups, stable cohomotopy group...