SIMS PhD Theses

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Browsing SIMS PhD Theses by Author "Otieno, Paul Antony"

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    Determining the rational homotopy type of the component of inclusion in the space of continuous mappings from gr(k, n) to gr(k , n + r)
    (Strathmore University, 2020) Otieno, Paul Antony
    The complex Grassmann manifold Gr(k, n) is the space of k dimensional sub spaces of en. Fork= 1, one gets epn-l, the space of lines in en. There is a natural embedding G(k , n) <-+ G(k, n + r). Moreover, any complex manifold can be embedded in some projective space epN. In particular, there is an embedding Gr(k, n) <-+ epN-l where N = (~) 0 To a simply connected topological space, Sullivan associates in a functional way a commutative differential graded algebra (cdga) henceforth (/\ V, d) which encodes the rational homotopy type of X. This is called a Sullivan model of X . Given that H *(ePn, Q) is the truncated polynomial algebra 1\xj(xn+l ), one gets a Sullivan model of the form (1\(x, y), d) where lxl = 2, jyj = 2n + 1 and dx = 0, dy = xn+l . For k 2: 1, one might use the homeo morphism G(k, n) = U(n)/(U(k) x U(n- k)) to find a Sullivan model. Moreover, iff :X --+ Y is a continuous mapping between CW-complexes, then there is a commutative differential graded algebra ( cdga) morphism ¢ : (1\Vy, d) --+ (1\Vx, d) between Sullivan models of X andY. This is called a Sullivan model of f. In this thesis, we use a Sullivan model of the inclusion Gr(k, n) ~ Gr(k, n+ r) to compute the rational homotopy type of the component of the inclusion in the space of mappings from Gr(k , n) to Gr(k, n + r). Further, we will compute an L00-model of the component of the inclusion i and deduce its Sullivan model, using the generalised cochain Quillen functor. We seek to define a model both Sullivan and Quillen for the component of the inclusion and from it obtain the cohomology algebra and even attempt to determine whether the space is formal or not.